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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
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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 The pseudopotential matrix element (|T|2) has been replaced by the structure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

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
଴

௄

ଶ௞ಷ

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௜௄.ሺோିோ೔ሻሽ ௗோ

… … . ሺ 4ሻ
37


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


ܷ 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௝ ൅ ∑௜ஷ௝ஷ௞ ‫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

௠ሺଶ௠ாሻ ଶ

ଵ

ቀఆቁ ∑ଵሺ2݈ ൅ 1ሻ sin ߟଵ ሺ‫ ܧ‬ሻ݁ ௜ఎభ ሺாሻ ௉భሺ௖௢௦ఏሻ

… … 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

ଶ௞ಷ

… . ሺ16ሻ

ߩൌቆ ଶ
ቇ න |ܶ|ଶ ‫ ݍ‬ଷ ݀‫ݍ‬
ଶћ ‫ ݒ‬ଶ݇ ସ
݁
ி
ி
଴

39


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 ݀‫ ݁ ݐ‬௜௪௧ ൏ ෍ ݁ ି௜௞.ோ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

∞

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
‫݁ܨ‬ଵ଴଴ି௫ ‫ܤ‬௫ , ‫଼݁ܨ‬଴ ‫݁ܤ‬ଶ଴ି௫ ܵ݅௫ , ሺ‫݁ܨ‬ଵ଴଴ି௫ ‫ܥ‬௢௫ ሻ଻ହ ‫ܤ‬ଶହ , ሺ‫݁ܨ‬ଵ଴଴ି௫ ‫ܥ‬௢௫ ሻ଼଴ ‫ܤ‬ଶ଴ , ሺ‫݁ܨ‬ଵ଴଴ି௫ ܰ݅௫ ሻ଻ହ ‫ܤ‬ଶହ ,
ሺ‫݁ܨ‬ଵ଴଴ି௫ ܰ݅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


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|௏௔ሺଶ௄ಷ ሻ

ழ|௏ሺ௄ሻ|మ ௔ሺ௄ሻவ

െ

ଵ
ଶ

… … ሺ25ሻ

ങ|ೇሺ಼ሻ|మ
வ
ങ಼ಷ
మ ௔ሺ௄ሻவ
ழ|௏ሺ௄ሻ|

ழ௄ಷ

… … 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


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


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.

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