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1. International Journal JOURNAL OF ADVANCED RESEARCH Technology (IJARET), INTERNATIONAL of Advanced Research in Engineering and IN ENGINEERING ISSN 0976 – 6480(Print), ISSNAND – 6499(Online) Volume 5, Issue 1, January (2014), © IAEME 0976 TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 5, Issue 1, January (2014), pp. 36-44 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) www.jifactor.com IJARET ©IAEME A NEW FORM OF EXTENDED ZIMAN-FABER THEORY FOR LIQUID AND AMORPHOUS BINARY ALLOYS K. Singh*, Brajraj Singh and R. Chaudhary Faculty of Engineering and Technology, Mody Institute of Technology and Science, Lakshmangarh, Rajasthan, India * Department of Physics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana ABSTRACT A new and simple form of the extended Ziman-Faber theory for liquid and amorphous metal has been worked out. The pseudopotential matrix element (|T|2) has been replaced by the structure factor and the atomic form factor. The modified formula has been used to calculate the electrical resistivity and the diffusion thermopower for Fe2B metallic glass. The results obtained from this theory give strong indication that the modified form of the extended Ziman-Faber theory can be applied successfully for similar systems. Fairly good agreement is obtained with the known experimental and theoretical results. INTRODUCTION The study of the electrical transport properties of metallic alloys has received a considerable attention both theoretically and experimentally. A large number of experimental and theoretical investigations have been reported aiming to understand mainly the temperature dependence of these properties and the effect of composition change. Much of the success of these studies has resulted from the application of the Ziman pseudopotential formulation for these systems [1,2,3]. Modified form of Ziman’s theory had been applied to some metals notably to the simple metals [4]. The phenomenon of thermoelectricity acquired prominence in the technological field as a result of its application to refrigeration. This particular field is dominated by the semiconducting materials which have sufficiently large thermo-electrical coefficients. Thermoelectric studies have also significantly advanced our understanding of metals, especially their electronic structures and scattering process [5]. Metallic glasses are classified under amorphous metallic alloys. The later may be divided into two types of structures. The first state is the microcrystalline state which occurs when bonds between atoms have a metallic or ionic character and the second is the vitreous state when bonds are of the 36
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 1, January (2014), © IAEME covalent type. In this case one of the constituents is said to be the glass former, such as phosphorus in Ni-P alloy or Silicon in Pd-Si alloy. Numerous attempts had been made to describe the short range order existing in the materials. From X-ray studies it had been found that local order vanished at distances of the order of 10 to 20A0[6]. This short range order suggests considerable degree of cystallinity in amorphous systems. Forgurassy et.al.[7] studied the effect of composition change for a group of materials based on the Fe-B alloy system. They tried to interpret the trends with composition in the amorphous state by using a modified version of the extended Ziman-Faber theory for the transport properties. By studying the crystallization they looked for correlation between the alloying elements added to the Fe-B system and the resulting crystalline phases. In the present work a modified form of the extended Ziman-Faber theory for the electrical transport properties of metals is employed to calculate the electrical resistivity and thermo-power of a binary alloy Fe2B. The calculated value of the electrical resistivity and thermopower of the proposed alloy system is compared with theoretical and experimental results [7]. Ziman-Faber theory of resistivity of binary alloys The electrical resistivity of a solid binary alloy is governed by two well known rules. The first rule, suggested by Nordhem [8], says that, as a function of atomic concentration(C), the resistivity ρ should be roughly proportional to C (1-C). The second rule suggested by Linde [9] says that if there is a difference between the valency of the solute (z1) and of the solvent (z0), the dρ/dC of the dilute alloy is approximately proportional to (z1 – z0)2. The fact that these rules are not generally true for liquid alloys seems to have attracted little attention even though a variety of liquid systems have been investigated experimentally. When the solvent is a polyvalent metal such as lead or tin these rules do not apply. It is only when the solvent is a monovalent metal such as copper that the resistivity of the liquid binary alloy behaves in the same sort of way as in the solid. The resistivity of a liquid metal takes the simple form: ߩൌቀ ଷగ ћ మ ௩ಷ మ ଵ ቁ ܷ݈ ሺܭሻ݈ ଶ 4 ቀ ଶಷ ଷ ቁ ቀ݀ܭൗ2݇ ቁ ி … . ሺ1ሻ This may be written as ߩ ൌ ቀ3ߨൗћ݁ ଶ ݒଶ ܸ ቁ ܷ|ۃሺܭሻ|ଶ ۄ ி … . ሺ2ሻ Where brackets, thus < > define an average over range of K from 0 to 2KF. Here V is the total volume of the specimen, vF is the Fermi velocity, K is the scattering vector and U (K) denotes a Fourier component of U(R): ܷሺ ܭሻ ൌ ܷ ሺܴ ሻ ݁ ሺ.ோሻௗோ … … . ሺ3ሻ The integral being carried over the whole volume. In a pure metal where all the ions are of the same species, the cores of the adjacent ions do not overlap, U(R) may be calculated as the superposition of screened pseudopotentials for each ion of the from Ui(R - Ri) where Ri denotes the centre of the ith ion, and if ܷ ሺ ܭሻ ൌ ܷ ሺܴ െ ܴ ሻ ݁ ሼ.ሺோିோሻሽ ௗோ … … . ሺ 4ሻ 37
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 1, January (2014), © IAEME it follows that ܷ ሺܭሻ ൌ ∑ ܷ ݁ ሺ.ோሻ … … . ሺ5ሻ Assuming Ui(R - Ri) is spherically symmetrical then Ui(K) is always real and independent of the direction of K. In a pure metal where all the ions are of the same species, we have మ തതതതതതതതതതത ൌ ܷ ሺܭሻଶ തതതതതതതതതതതതതതതത |ܷ ሺܭሻ|ଶ ห∑ప ݁ ሺ ప .ோഢ ሻ ห ൌ ܰ ܷ ሺܭሻଶ ܽሺܭሻ so that ߩ ൌ ቀ3ߨܰൗћ݁ ଶ ݒଶ ܸ ቁ ܷ|ۃଶ ܽሺܭሻ|ଶ ۄ ி … … ሺ 6ሻ In this way the resistivity is determined by a product of a function, U2 which is a property of the ions alone, and the interference factor, a(K) which is, strictly speaking, an ensemble depending on the structure. For a pure metal the interference function by an ensemble average can be evaluated as follows: തതതതതതതതതതതതതതതതതതതത ܽሺܭሻ ൌ ܰ ିଵ ∑ప ∑ఫ ݁ ప.൫ோഢ ି ோണ ൯ തതതതതതതതതതതതതതതതതതതതതത ൌ 1 ܰ ିଵ ∑ప ∑ఫஷప ݁ ప.൫ோഢ ି ோണ ൯ ே ܽ ሺܭሻ ൌ 1 ሼܲሺݎሻ െ 1ሽ ୱ୧୬ … . . ሺ7ሻ 4ߨ ݎଶ ݀ݎ To deal with alloys it is convenient to introduce a number of analogous functions aαβ (α and β are treated as dummy suffices which run over all the different species of ion which may be present in the alloy). The average distribution of α ions round a given β ion can be described by a paircorrelation function Pαβ (r), i.e. Pαβ is the probability of finding an α ions per unit volume at a radius r from the centre of a β ion, normalised in such a way it tends to unity for large r. We set ܽఈఉ ൌ 1 ே ൛ܲఈఉ ሺݎሻ െ 1ൟ ∞ ୱ୧୬ … . . ሺ8ሻ 4ߨ ݎଶ ݀ݎ Defined in this way, aαβ is independent of specimen size and is unity for a completely random gas-like structure. ݁ .൫ோഀ ି ோഁ ൯ ൌ ߜ൫ܴఈ െ ܴఉ ൯ ݁ .൫ோഀି ோഁ൯ ൌ ܰܥఈ ߜఈఉ ܰܥఈ ܥఉ ൫ܽఈఉ െ 1൯ … … … . ሺ9ሻ where Cα and Cβ are the concentrations of the 2 species: the initial δ – function is needed to cover the case where α and β happen to describe the same species, so that Rα and Rβ can refer to the same ion. N is the number of ions per unit volume. Thus for an alloy [4] 38
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 1, January (2014), © IAEME ܷ ሺܭሻଶ ൌ ∑ఈ ∑ఉ ܷఈ ܷఉ ൛ܰܥఈ ߜఈఉ ܰܥఈ ܥఉ ൫ܽఈఉ െ 1൯ൟ ൌ ∑ఈ ܰܥఈ ܷఈ ଶ െ ∑ఈ ∑ఉ ܰܥఈ ܥఉ ܷఉ ܷఉ ∑ఈ ∑ఉ ܰܥఈ ߜఈఉ ܷఈ ܷఉ ܽఈఉ തതതത തതതത ൌ ܰ൛ܷ ଶ ܷ ଶ ൟ ∑ఈ ∑ఉ ܥఈ ܥఉ ܷఈ ܷఉ ܽఈఉ … … ሺ10ሻ … … ሺ11ሻ EXTENDED FABER-ZIMAN (F-Z) THEORY FOR BINARY ALLOYS Nearly Free electron(NFE) Model in terms of the T – matrix In the NFE model, the scattering of electrons are assumed to be elastic and the electron states are plane waves |K> and |Kl > [5]. A transition operator, the T – matrix describes the scattering of an electron through a system of N scatterers. The operator is so defined that its matrix elements give the probability of scattering from state |K > to |Kl>. The T- matrix can be written in the form [10] ܶ ൌ ∑ ݐ ∑ஷ ݐ ܩ ݐ ∑ஷஷ ݐ ܩ ݐ ݐ … … ሺ12ሻ where GO is the propagator of a free particle, ti is the transition matrix of single scattering centre at site i and repeated scattering at the same centre are not allowed. The T- matrix gives the probability of an electron in state |K> being scattered into |Kl> by a single scattering event. For scattering in the energy shell the T matrix can be written ݐሺ݇, ݇ ூ ሻ ൌ ିଶగћ ଵ ሺଶாሻ ଶ ଵ ቀఆቁ ∑ଵሺ2݈ 1ሻ sin ߟଵ ሺ ܧሻ݁ ఎభ ሺாሻ భሺ௦ఏሻ … … ሺ13ሻ where η1(E) is the phase shift of partial wave of orbital angular momentum l at energy E, m is the mass of electron and P is the legendary polynomial of order l. For the binary alloy problem, following the analysis of Faber and Ziman [4], the resistivity can be calculated using the following: ߩൌቀ ଷగఆை మ మ ћ ௩ಷ మ ଷ ଵ ݍ ݍ ቁ ܽ ሺ ݍሻ|ݐሺ݇, ݇ ூ ሻ|ଶ 4 ቀ ൗ2݇ ቁ ݀ ቀ ൗ2݇ ቁ ி ி … . ሺ14ሻ Where q = k – k’, is the momentum transfer in the process of scattering. Hence for a binary alloy ଶ ܽሺ ݍሻห݇ , ݇ ′ ห is replaced by: |ܶ|௩ ଶ ൌ ܿଵ ݐଵ ଶ ሾ1 െ ܿଵ ܿଵ ܽଵଵ ሺݍሻሿ ሾܿଶ ݐଶ ଶ ሿሾ1 െ ܿଶ ܽଶଶ ሺ ݍሻܿଶ ሿ ܿଵ ܿଶ ሾݐଵ ݐ כଶ ݐଶ ݐ כଵ ሿሺܽଵଶ ሺݍሻ െ 1ሻ … … ሺ15ሻ where c1, c2 are the atomic concentrations of components 1 and 2 which have t – matrices t1 and t2 respectively and a11 (q), a22 (q) and a12 (q) are the interference functions. Hence a modified expression for the resistivity is given by 3ߨߗܱ ଶಷ … . ሺ16ሻ ߩൌቆ ଶ ቇ න |ܶ|ଶ ݍଷ ݀ݍ ଶћ ݒଶ݇ ସ ݁ ி ி 39
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 1, January (2014), © IAEME NEW FORM OF EXTENDED F-Z THEORY A new form of F-Z theory has been worked out using lattice dynamical approach. The F-Z theory relates the resistivity to a simple integral over the liquid structure factor, giving information about the ion position, and the square of the electron – ion pseudopotential form factor, describing electron – ion scattering. Bayin [11] and Ziman [12] have given a formulation of the ideal resistivity of a solid which closely parallels the work done in liquids, and involves the dynamical structure factor for the ion system. At sufficiently low temperature it can be related with good accuracy to the phonon spectrum using the one phonon approximation. Dyes and Carbotte [13] obtained an expression for the resistivity of simple metals giving by ߩ ൌ ܿ න ݇݀ଷ ݇ |ܹሺ݇ሻ|ଶ න ݀ܵ ݓሺ݇; ݓሻ ߚݓ ݁ఉ௪ െ 1 … … ሺ17ሻ Where c is a constant, W (k) is the pseudopotenial form factor, and S (k;w) is the space-time Fourier transform of the dynamical structure factor. The pseudopotential describing electron scattering at the Fermi surface is assumed to depend only on momentum transfer k. Further, the Fermi surface is taken to be spherical so that the two surface integrals describing transitions form an initial state to a final state on the Fermi surface can be converted to three dimensional integral over k. The phonon frequencies and polarisation vectors are completely determined in the first Brillomin zone (FBZ) from the force constants of the material. These describe the force on a given atom due to a displacement of another. The force constants can be obtained at least for the few nearest neighbour shells by a Born-ion Karman analysis of the experimental dispersion curves usually measured only in the high symmetry directions. The force constant so obtained, however can be used to determine the dynamical matrix at any point in the FBZ and consequently the lattice dynamics [13]. Dynamical Structure Factor The structure factor S (k; w) describing the ion system is perfectly general and can be expressed as ∞ ܵሺ݇ ݓሻ ൌ න ݀ ݁ ݐ௪௧ ൏ ݁ ି.ோభሺ௧ሻ ݁ ..ோభሺ௧ሻ ݐ … … ሺ18ሻ ଵ,ଵ ି∞ where the brackets means a thermal average, R1 (t) is the instantaneous position of the 1th ion at the time t, and the sums extended over all ions. The structure factor as defined provides a direct measure of the density fluctuations of the lattice. All multiple-phonon processes are included. At sufficiently low temperatures, phonons are well-defined elementary excitations of the ions. Also one phonon approximation should be accurate. Under their conditions S (kw) can be written as ܵሺ݇ ; ݓሻ ൌ గே ெ ∑ఒ |.ఌሺ ; ఒሻఘ| ௪ሺ ; ఒሻ ቄ ఋ൫௪ା௪ሺ ; ఒሻ൯ ഁೢሺೖ ; ഊሻ ିଵ ఋ൫௪ି௪ሺ ; ఒሻ൯ ଵି షഁೢሺೖ ; ഊሻ ቅ … … . ሺ19ሻ The sum extended over the FBZ, w (k ; λ) is the phonon frequency and ԑ (k ; λ ) is the polarisation vector. M is the mass of Fe2B molecule. It can be shown that 40
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 1, January (2014), © IAEME ∞ න ݀ݏݓሺ݇ ; ݓሻ ି∞ |݇. ߝሺ݇ ; ߣሻ|ଶ ߚݓ 2ߨܰ ൌ ఉ௪ћ ሺ݁ ܯ ݁ఉ௪ െ 1 െ 1ሻሺ1 െ ݁ ିఉ௪ћ ሻ So that ߩሺܶሻ ൌ ܿ ூ න ݇݀ଷ ݇ |ܹሺ݇ሻ|ଶ ఒ |݇. ߝ ሺ݇ ; ߣሻ|ଶ ܭ ܶ ሺ݁ఉ௪ћ െ 1ሻሺ1 െ ݁ ିఉ௪ћ ሻ … … ሺ20ሻ … … ሺ21ሻ The constant c1 is given by ܿூ ൌ Where o 3݄ߗܱ … … ሺ22ሻ ݁ܯଶ ݒி ଶ 16݇ி ସ is the volume per ion. Hence ߩሺܶሻ ൌ න ܿ ூ ݇݀ଷ ݇ ܵሺ݇ሻ |ܹሺ݇ሻ|ଶ ܵ ሺ݇ ሻ ൌ … … ሺ23ሻ |.ఌሺ ; ఒሻ|మ ⁄ಳ ் ൫ ഁೢћ ିଵ൯൫ଵି షഁೢћ ൯ Taking the form factor W (k) to depend only on the magnitude of the momentum transfer, the angular integration in Eqn (23) averages S (k) and the formula is identical to that for liquids. It is a simple integral over momentum k between 0 and 2 KF of overlap of the square of the pseudopotential and the spherically averaged structure factor. DETERMINATION OF DISPERSION RELATION Possible Model of Vibration for Fe2B The most important thing to know before applying lattice dynamical approach on a particular system is the structure and therefore the arrangement of atoms or ions in the system. However, very little information about the structure of metallic glasses is known. Where there exists acknowledge about the corresponding crystalline forms one utilises the general rule that in the amorphous systems the nearest neighbour configuration is probably the same. Fograssy et. al.[7] studied a group of materials based on the Fe-B system changing systematically both the metallic and the metalloid components. The electrical resistivity and the thermopower of ݁ܨଵି௫ ܤ௫ , ଼݁ܨ ݁ܤଶି௫ ܵ݅௫ , ሺ݁ܨଵି௫ ܥ௫ ሻହ ܤଶହ , ሺ݁ܨଵି௫ ܥ௫ ሻ଼ ܤଶ , ሺ݁ܨଵି௫ ܰ݅௫ ሻହ ܤଶହ , ሺ݁ܨଵି௫ ܰ݅௫ ሻ଼ହ ܤଵହ , ଼݁ܨ ܶܯଷ ܤଵ amorphous alloys were measured from room temperature to the crystalline transition for different compositions. In the last group of samples TM stands for one of the 3d4d and 5d elements. In this study of the Fe2B amorphous system, the chain like structure is considered. Two possible structures with the same atomic vibration are considered and the dispersion relations for these systems are determined. 1. In the first structure of Fe2B units are assumed to form a one dimensional chain in which tow atoms are ironically bounded to a boron atom. 2. In the second structure diatomic molecules of iron are alternately bounded to single boron atoms. In both models only central forces are considered. 41
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 1, January (2014), © IAEME Diffusion Thermopower The diffusion thermopower for a free-electron system obeying Fermi-Dirac statistics in which a single relaxation time exists can be expressed as మ మ ଷ … … ሺ24ሻ ܵ ൌ െ ሺଷ⁄ሻಳ ቀଶ ݉ቁ ൗா ಷ The evaluation of the thermopower within the Ziman frame work follow quite simple once the sensitivity expression is obtained. The thermopower is then given by ܵ ൌ െ Where ߛൌ ߨ ଶ ܭ ଶ ܶߛ 3 ൬ ݉൰ ሺ 3⁄ ݁ ሻ ൗ2 ܧ ி ଷିଶ|ሺଶಷ ሻ ழ|ሺሻ|మ ሺሻவ െ ଵ ଶ … … ሺ25ሻ ങ|ೇሺ಼ሻ|మ வ ങ಼ಷ మ ሺሻவ ழ|ሺሻ| ழಷ … … ሺ26ሻ Electrical Resistivity The modified form of the extended Ziman-Faber theory for the electrical resistivity of liquid and amorphous metals is giving by … … ሺ27ሻ ߩ ൌ ܵሺܭሻ |ܸሺܭሻ|ଶ ܭଷ ݀ܭ Where the structure factor S (k) takes the form ܵሺ݇ ሻ ൌ |݇. ߝ ሺ݇ ; ߣሻ|ଶ⁄ܭ ܶ ሺ݁ఉ௪ћ െ 1ሻሺ1 െ ݁ ିఉ௪ћ ሻ … … ሺ28ሻ The atomic form factor is given by ܸ ሺ݇ ሻ ൌ െ 4ߨ ܼ݁ ଶ ߗ ݇ଶ … … . ሺ29ሻ The structure factor was evaluated at each momentum wave vector using the dispersion relation. The polarisation vector ԑ ( q ; λ ), was taken to be unity for one dimensional lattice. The atomic form factor depends only on the momentum transfer k. this formula was obtained by Fourier transforming the single ionic pseudopotential, given by ଶ ܸሺݎሻ ൌ െܼ݁ ൗݎ … … ሺ30ሻ Where r is the ionic radius. Using this formula, the result obtained for the sensitivity of Fe2B amorphous system agreed fairly well with experimental and other theoretical results. This affirms the validity in applying this formula to these systems. A typical data is show in Table 1 and Table 2 42
8.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 1, January (2014), © IAEME Table 1 A typical data set for amorphous Fe2B Input Data Atomic number Z=5 Atomic weight of Fe Afe = 55.847 gm Atomic weight of B AB = 10.81 gm Boltzmann constant KB = 1.38 x 10-23 JK-1 Electronic charge e = 1.6 x 10-19 C Planck’s constant h = 6.63 x 10-34 Js Lattice constant of B a = 1.82 A0 Velocity of sound through Vo = 5 x 103 ms-1 glass Atomic radius of B rB = 0.91 A0 Atomic radius of Fe rFe = 1.26 A0 Calculated data: Table 2 Mass of Fe2B molecule Atomic volume m = 11.072 x 10-26 kg o = 23.06 x 10-30 m3 Fermi wave vector KF = 1.725 A-1 Absolute mass of Fe M1 = 9.276 x 10-26 kg Absolute mass of B M2 = 1.796 x 10-26 kg Mass of Fe2B molecule M = M1+M2 Wave Vector K/ A-1 0.345 0.431 0.575 0.863 1.150 1.725 T = 300 K Theoretical Published Experimental Frequency w/1013 0.870 1.111 1.451 2.052 2.213 2.902 Electrical Resistivity /µ cm 128.8 129.5 133.0 Thermopower /µVK-1 -2.6 -2.6 -3.1 Reference This work (7) (7) CONCLUSION The pseudopotentials matrix element |T|2 has been replaced successfully by the structure factor and the atomic form factor. The results obtained for Fe2B alloy using the new form of extended Faber-Ziman theory for electrical resistivity and thermo-power compare well with experimental as well as the published results. 43
9.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 1, January (2014), © IAEME REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. Ziman, J. M. , Adv. Phys. 16, ( 1967 ), 55 Meyer, A. , Nestor, J.C.W., Young, W. H., Adv. Phys. 16, (1967), 581. Ziman, J. M. , Phil. Mag. 6, (1961), 1013. Faber, T. E. and Ziman, J. M. Phil. Mag. 11, (1965), 153. Barnard, R.D., Thermoelectricity in metals and Alloys, University of Salford, London, (1972), 58. Enderly, J. E. and Simons, C.J., Phil. Mag., 20, (1969), 125. Forgarassy, B., Vasvari, B., Szab, I., Jafar, E., Electrical Transport properties of (FecTM1-c)B1-n type Metallic Glasses, Hungary Academy of Sciences, Central Institute for Physics, Budapest (1980). Nordheim, L., Ann. Phys. Lpz., 9, (1931), 641. Linde, J. O. Ann. Phys. Lpz., 15, (1932), 239. Deeby, J. L. And Edwards, S. F., Proc. Royal Soc., A274, (1963), 395. Baym, G., Phys. Rev., 135, (1964), 1691. Ziman, J. M., Electrons and Phonons, Clarenden Press, Oxford, England, (1960). Dynes, R. C. And Carbotte, J. P., Phys. Rev., 175, (1972), 687. Brajraj Singh, D C Gupta, R. Chaudhary, K.Singh and Y M Gupta, “Comparative Study of Layered Structure Formulisms for High Temperature Copper-Oxide Superconductors”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 2, 2013, pp. 46 - 60, ISSN Print: 0976-6480, ISSN Online: 0976-6499. Brajraj Singh, D C Gupta, R. Chaudhary, K.Singh and Y M Gupta, “Strong Coupling Model for High-Tc Copper-Oxide Superconductors”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 1, 2013, pp. 134 - 141, ISSN Print: 0976-6480, ISSN Online: 0976-6499. T. Opoku-Donkor, R. Y. Tamakloe, R. K. Nkum and K. Singh, “Effect of Cod on OCV, Power Production and Coulombic Efficiency of Single-Chambered Microbial Fuel Cells”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 7, 2013, pp. 198 - 206, ISSN Print: 0976-6480, ISSN Online: 0976-6499. 44
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