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CHAPTER-7 284

Punit H Suthar/Ph.D (Physics)/Sardar Patel University/May-2012

CHAPTER-7

Total Energy and Equation of state for

Solids solutions

“Total energy of some FCC metals” 22nd Gujarat Science Congress (GSC) organized by Bhavnagar University,

Bhavnagar.

7.1 Introduction 285

7.2 Theory and Method of computation 287

7.3 Results and Discussions

7.3.1 Total energy

7.3.2 Pressure and Volume relation

7.3.3 Energy and Volume relation

291

291

298

305

References 312

CHAPTER-7 285


7.1 Introduction

It is known that the electronic properties of some metals and alkaline earth

metals, some transition metals and their alloys can be understood using the

pseudopotential theory. The total energy of pure metals in the framework of the

second order perturbation theory has been a subjected of many pseudopotential

studies [7.1-715]. However the underlying assumption in the calculation of the total

energy of alloys is the same as that for pure metals. Such studies on the binary alloys

are very limited [7.16-7.24].

The equation of state (EOS) is the relationship between pressure, volume and

temperature, i.e. the (P, V, T) relation. They are of immense importance to

theoreticians as well as experimentalists. In theoretical physics, they provide a test to

the theoretical models of cohesion and predict the onset of phase transition (insulator

to metal, valence transition, solid to liquid, stable crystal structure, etc.). They are also

used for pressure calibration in high-pressure experiments and are needed to relate the

measured pressure variation of some physical quantity (e.g. elastic constants,

transport properties, specific heat, etc.) to the calculated volume variation. It has been

discovered that d-electrons (Cr, Ni, Cu) play a crucial role in determining the stable

structure of not only the transition metals but also of the alkaline- earth and group II-

B metals. For the transition metals from left to right across the periodic table, the

trend seems to be bcc and fcc sequence of structures [7.15] equation of state, total

crystal energies, in the case of alkaline earth metals, the trend explained by the

presence of d-electrons and f-electrons includes the fcc and bcc sequence observed

with increasing atomic numbers [7.25, 7.26] that can be studied using energy–volume

relation and equation of state. In astrophysics the, EOS is used to unravel the

mysteries of evolution of stellar bodies like white dwarfs, neutron stars and black

holes while in geophysics it helps to understand the structure of the Earth. The range

of the pressure interior of the Earth is 350 GPa. Various experimental techniques were

used to study the EOS under high pressure. The data by Bridgman [7.27] using the

piston cylinder method were the first to be reported and the pressure range was about

5–10 GPa (static). In a similar way Vaidya and Kennedy [7.28] made careful

measurement up to about 4.5 GPa. Drickmer [7.29] extended the pressure range up to

CHAPTER-7 286


20–30 GPa using a supported anvil device. In the early seventies, the range of static

pressure EOS has been extended to over 100 GPa using diamond anvil devices [7.30-

7.31]. The development of dynamic shock wave techniques using chemical explosives

increased the pressure range to 50 GPa [7.32, 7.33]. Since then a number of

techniques (e.g. lasers [7.34], electron guns [7.35], and underground nuclear

explosions [7.36]) have been developed for generating controlled dynamic high

pressures, whose limit is 10 GPa. While studying EOS theoretically, the major

difficulty arises from the question how to incorporate correctly the structurally

complicated inter-particle interactions of the many-body problem. Previously, many

authors [7.19, 7.20, 7.37–7.45] have studied the EOS using different techniques such

as the tight-binding total-energy classical cell model [7.41], the generalized

pseudopotential theory [7.42], Debye-Gr¨uneisen theory [7.43], recent classical mean-

field model [7.44], linearised augmented plane-wave method with both the local

density and generalized gradient approximation [7.45] and model pseudopotential

methods [7.37–7.40]. In the present work, we have studied the EOS for all alkaline

earth metals, some transition metals and their binary alloys using two different

pseudopotentials. The EOS of Cu has been studied up to a pressure of 1000 GPa, for

Ni the pressure range is up to 55 GPa. We have also calculated the total energy,

pressure - volume relation and energy- volume elements. We calculate some atomic

properties of Ba, Cr (BCC), Al, Ca, Ni, Cu, Sr (FCC) metals, and NiAl, Ni3Al and

Al 2Sr binary system using the pseudopotential theory and second order perturbation

theory. We have selected this particular metals and alloys, just because of their

extraordinary properties and many applications in the industry. It is known that the

intermetallic compounds like NiAl [7.46], Ni3Al [7.46] and Al2Sr[7.47] demonstrates

a number of attractive properties that motivate its extensive use in industry [7.1]. The

alloys of Al with transition metals show advantageous properties such as high

strength, lightness, corrosion resistance, recyclability and formability. Hence

aluminum-transition metal alloys are being employed in ever-increasing number of

applications as in ballast, step soldering, radiation shielding, alloy casting, in

construction of modern aircrafts and rockets, to coat telescope mirrors, packaging,

architecture and electrical transmission applications [7.1, 7.48].

The atomic properties like total crystal energy, pressure, energy – volume

relations and pressure – volume relations are computed for Al, Ca, Ni, Cu, Sr (FCC),

CHAPTER-7 287


Ba, Cr (BCC) and their binary (NiAl, Ni3Al, Al 2Sr) solid solutions using the

reciprocal sum analysis method. The well tested model potentials [7.49, 7.50] along

with six different forms of local field correction functions (LFCF) namely Hartree

[7.3], Taylor [7.51], Ichimaru-Utsumi [7.52], Farid et al. [7.53], Sarkar and Sen [7.54]

and Hubbard Sham [7.55, 7.56] local field correction functions to compute the

screened form factors, which are then used to calculate the equation of state for many

solid solutions of interest.

7.2 Theory and Method of computation

It is difficult to compute static and dynamic properties of crystals such as total

energy, pressure, bulk modulus and lattice vibrations using information obtained from

the band structure calculation. Second-order perturbation theory based on the

pseudopotential formalism is used to study the properties of crystals.

The total energy of a crystal may be regarded as the sum of two contributions,

one of which depends only on the volume of the crystal, and the other, formally, only

on the crystal structure. The former includes terms which are independent of the

atomic distribution, namely, the kinetic energy of free electrons, the exchange and

correlation energies. The structure dependent energy is the sum of electrostatic energy

and the band energy. The total crystal energy per atom of pure metal has been

computed by many workers using pseudopotential calculations upto the third order in

the perturbation approach [7.57].

The first work on the band energy of alloy was done by Hayes and co-

workers [7.58] using a non-local potentials. Gurskii and Krasko [7.59] have reported

the total binding energy for all the five alkali metals using a model potential which is

continous in r-space. Soma [7.5, 7.19, 7.20] was successful in computing total energy

and bulk modulus of some co-valent compounds on the basis of extended perturbation

theory. They have used historical model potentials and local Heine-Abarenkov model

potential [7.60 ] with an additional parameter to ensure minimum energy condition.

CHAPTER-7 288


Hafner [7.18] derived an orthogonalized plane wave based on the first

principles pseudopotential method and applied it to investigate the random binary

alloys and ordered intermetallic compounds between alkali metals. This treatment is

the virtual cryatal approximation in which the disordered alloy is replaced by a

monoatomic periodic lattice and the alloy potential is considered as a linear

combination of the average lattice potential and the difference potential. Tanigawa

and Doyama [7.61] have used a different approach for the study of the total crystal

energy of alloys. This approach is known as the pseudo alloy atom (PAA) model in

which a hypothetical monoatomic crystal is supposed to be composed of pseudo alloy

atoms. In this model the hypothetical crystal made up of pseudo alloy atoms is

supposed to have same properties as the actual disordered alloy crystal and the

pseudopotential formalism is then applied to calculate various properties of an alloy.

To compute the total crystal energy of the alloy in the present study, we have

used the pseudo alloy atom model proposed by Tanigwana and Doyama[7.61].

In the pseudo alloy atom model the alloy can be considered as composed of a

system of N periodically arranged positive ions immersed in an electron gas. The

alloy of the type A1-xBx is made up of two components A and B in which x is the

arbitrary concentration of element B. The mean valence Z alloy for such a solid

solution can be obtained from the relation,

Zalloy = (1-x) ZA+ x ZB (7.1)

Where ZA and ZB are the valence of the two components which make up the alloy.

The total crystal energy per pseudo-atom, E(x), of the binary alloy can be obtained in

the framework of the usual second order perturbation as [7.5, 7.19-7.24]

E (x) = Ei (x) + Ees (x) + E1 (x) + E2 (x) (7.2)

In equation (7.2) , Ei is the electrostatic energy of point ions immersed in the uniform

gas of valence electrons, called the Madelung energy , which is given by

CHAPTER-7 289


a

alloyi R

ZxE

2

)(α

−= (7.3)

Here α is the Madelung’s constant and R a = rs Z1/3

alloy, Here rs is the radius of the

sphere containing one electron.

In Equation (7.2) ,Ees is the sum of the kinetic, exchange and correlation energies of

the valence electron and is given by

( )

+−−= s

ssalloyes r

rrZxE ln031.0115.0

916.021.2)(

2 (7.4)

E1 is the First order perturbation energy of the valence electron due to the

pseudopotential and is given by,

+

Ω=

→)(

4lim)(

20

22

01 qWZ

q

eZxE Balloy

alloy

alloy

q

π (7.5)

and

∑≠

=0

2 )()(q

qFxE (7.6)

is the band structure energy. Here F(q) is the normalized energy wave number

characteristics and is given by

[ ][ ][ ] )(1 1)(1

1)()(

16)(

22

0

qfq

qqV

qqF

effeffH

effHeff

Beff

eff −−+−

×Ω

−=ε

επ

(7.7)

The values of atomic volume and potential parameter rc for the alloy can be found

from the relation

BA xxalloy

Ω+Ω−=Ω )1(0 (7.8)

CHAPTER-7 290


cBcAc rrxralloy

+−= )1( (7.9)

Where AΩ and BΩ are the atomic volumes of the two pure components. rcA and rcB

are the parameters of the potential for the two components found by satisfying the

zero pressure condition.

The pressure P is obtained from the first derivative of the total crystal

energy per atom E with respect to the atomic volume 0Ω given by,

0Ω−=

d

dEP T

T (7.10)

Ω+

Ω+

Ω+

Ω−=∴

0

2

0

1

00 d

dE

d

dE

d

dE

d

dEP esi

tot (7.11)

We can write the above equation as

210 PPPPP i +++= (7.12)

By setting the derivative

00

=−=Ω

Pd

dE (7.13)

The total crystal energy at zero pressure of alkaline earth metals, transition metals and

their alloys are found using Equation (7.2) to Equation (7.6).

CHAPTER-7 291


7.3 Results and Discussions

Table-7.1: Input parameter and constant for the Solid solutions in present

computation

7.3.1 Total energy

The total crystal energy at zero pressure condition of some pure BCC (Ba, Cr)

and FCC (Al, Ca, Ni, Cu, Sr) and combination of their alloys in three different

compositions NiAl, Ni3Al and Al2Sr has been computed using two different model

potentials along with six different local field correction functions. The individual

terms contributing to the total crystal energy at zero pressure per atom are tabulated in

the tables from Table 7.2 to Table 7.11.

Metals Z Ω0 (a.u)3

Model rc (a.u)

H T IU F S HS

M-1 0.63030 0.59402 0.61312 0.61097 0.62588 0.61941 Al 3 111.3

M-2 0.69029 0.68983 0.67596 0.67591 0.67320 0.69055

M-1 1.04585 0.99951 0.99725 0.99086 1.02337 1.03063 Ca 2 293.492

M-2 0.97901 0.97373 0.99819 0.99817 0.99814 0.97797

M-1 0.58066 0.58093 0.57968 0.57823 0.57979 0.58527 Cr 1 81.012

M-2 0.57517 0.57708 0.57629 0.57643 0.57574 0.57580

M-1 0.55312 0.54732 0.55116 0.55258 0.55173 0.54845 Ni 1 73.793

M-2 0.54844 0.55078 0.54923 0.54931 0.54383 0.54922

M-1 0.58056 0.57464 0.57965 0.57833 0.57979 0.57947 Cu 1 79.715

M-2 0.57096 0.57312 0.57230 0.57240 0.57152 0.57169

M-1 1.17551 1.12414 1.11768 1.10981 1.15133 1.15869 Sr 2 380.442

M-2 1.12965 1.02616 1.09885 1.09879 1.10001 1.06776

M-1 1.23585 1.17661 1.13559 1.11856 1.19240 1.21751 Ba 2 428.515

M-2 1.11173 1.10309 1.12303 1.12284 1.12600 1.10938

M-1 0.58527 0.56063 0.57229 0.56956 0.58198 0.57845 NiAl 2 92.547

M-2 0.24820 0.62765 0.62230 0.62232 0.61911 0.62616

M-1 0.56551 0.54926 0.55788 0.55211 0.56281 0.56128 Ni3Al 1.5 83.17

M-2 0.58915 0.59695 0.58967 0.58731 0.58698 0.59053

M-1 0.77148 0.72482 0.74041 0.73716 0.75815 0.75576 Al 2Sr 2.8 165.13

M-2 0.80073 0.79757 0.79470 0.79472 0.79265 0.80013

CHAPTER-7 292


Table-7.2: Various contributions to the Total Crystal Energy (Ryd/atom) at zero

pressure for Aluminum (Al)

Variation contribution of energy (Ryd)

LFCF Model

(M) Ei E0 E1 E2 T.E.

%

error

Exp.

[7.27]

Others [7.62] [7.63], [7.64], [7.65]

M-1 -5.4040 -0.0567 0.9977 -0.2438 -4.7068 11.9 H

M-2 -5.4040 -0.0567 1.4525 -0.4370 -4.4453 6.81

M-1 -5.4040 -0.0567 0.8816 -0.3647 -4.9393 16.1 T

M-2 -5.4040 -0.0567 1.4506 -0.4606 -4.4707 7.34

M-1 -5.4040 -0.0567 0.9441 -0.2553 -4.7720 13.1 IU

M-2 -5.4040 -0.0567 1.3928 -0.0127 -4.0810 1.50

M-1 -5.4040 -0.0567 0.9374 -0.2593 -4.7826 13.3 F

M-2 -5.4040 -0.0567 1.3927 -0.0128 -4.0808 1.50

M-1 -5.4040 -0.0567 0.9837 -0.2046 -4.6816 11.5 S

M-2 -5.4040 -0.0567 1.3815 -0.0124 -4.0917 1.23

M-1 -5.4040 -0.0567 0.9636 -0.2817 -4.7789 13.3 HS

M-2 -5.4040 -0.0567 1.4537 -0.4422 -4.4493 6.89

-4.1423

-4.343,

-4.392,

-4.279,

-4.192,

-4.327


pressure for Calcium (Ca)

Variation contribution of energy (Ryd) LFCF

Model

(M) Ei E0 E1 E2

T.E. %

error Exp.

Others [7.66] [7.67]

M-1 -1.7384 -0.3035 0.4629 -0.0381 -1.6172 -- H

M-2 -1.7384 -0.3035 0.4924 -0.1352 -1.6848 --

M-1 -1.7384 -0.3035 0.4228 -0.0573 -1.6765 -- T

M-2 -1.7384 -0.3035 0.4872 -0.1426 -1.6975 --

M-1 -1.7384 -0.3035 0.4209 -0.0404 -1.6615 -- IU

M-2 -1.7384 -0.3035 0.5119 -0.0000 -1.5302 --

M-1 -1.7384 -0.3035 0.4159 -0.0426 -1.6690 -- F

M-2 -1.7384 -0.3035 0.5119 -0.0000 1.5301 --

M-1 -1.7384 -0.3035 0.4433 -0.2813 -1.6268 -- S

M-2 -1.7384 -0.3035 0.5119 -0.0000 -1.5301 --

M-1 -1.7384 -0.3035 0.4416 -0.0440 -1.6364 -- HS

M-2 -1.7384 -0.3035 0.4914 -0.9779 -1.6871 --

-- -1.3917

-1.3366

CHAPTER-7 293



pressure for Chromium (Cr)

Variation contribution of energy(Ryd) LFCF

Model


%

error

Exp.

[7.67]

Others [7.15] [7.68]

M-1 -0.6675 -0.1189 -0.2672 -0.0401 -1.3908 11.8 H

M-2 -0.6675 -0.1189 0.3533 -0.0475 -1.3120 6.57

M-1 -0.6675 -0.1189 0.2559 -0.0532 -1.4184 13.5 T

M-2 -0.6675 -0.1189 0.3572 -0.4980 -1.3105 8.48

M-1 -0.6675 -0.1189 0.2509 -0.0494 -1.4163 13.4 IU

M-2 -0.6675 -0.1189 0.3522 -0.0041 -1.2697 4.40

M-1 -0.6675 -0.1189 0.2478 -0.0511 -1.4213 13.7 F

M-2 -0.6675 -0.1189 0.3526 -0.0410 -1.2696 4.40

M-1 -0.6675 -0.1189 0.2549 -0.0436 -1.4065 12.8 S

M-2 -0.6675 -0.1189 0.3498 -0.0040 -1.2720 3.63

M-1 -0.6675 -0.1189 0.2633 -0.0437 -1.3983 12.3 HS

M-2 -0.6675 -0.1189 0.3547 -0.7932 -1.3111 6.51

-1.2257 -1.309,

-1.018


pressure for Nickel (Ni)

Variation contribution of energy (Ryd) LFCF

Model


%

error

Exp.

[7.67]

Others [7.15] [7.68] [7.69]

M-1 -1.5493 -0.0972 0.2621 -0.0450 -1.4295 4.77 H

M-2 -1.5493 -0.0972 0.3558 -0.0477 -1.3383 1.71

M-1 -1.5493 -0.0972 0.2472 -0.5988 -1.4592 6.71 T

M-2 -1.5493 -0.0972 0.3608 -0.5039 -1.3360 1.88

M-1 -1.5493 -0.0972 0.2562 -0.0466 -1.4368 5.26 IU

M-2 -1.5493 -0.0972 0.3557 -0.0068 -1.2976 4.90

M-1 -1.5493 -0.0972 0.2538 -0.0478 -1.4405 5.50 F

M-2 -1.5493 -0.0972 0.3558 -0.0060 -1.2976 4.90

M-1 -1.5493 -0.0972 0.2607 -0.0402 -1.4261 5.25 S

M-2 -1.5493 -0.0972 0.3521 -0.0066 -1.3010 4.62

M-1 -1.5493 -0.0972 0.2583 -0.0492 -1.4374 5.30 HS

M-2 -1.5493 -0.0972 0.3577 -0.0483 -1.3371 1.80

-1.3612

-1.373,

-1.214,

-1.074

CHAPTER-7 294



pressure for Copper (Cu)


Model


%

error

Exp.

[7.67]

Others [7.15] [7.68] [7.69]

M-1 -1.5099 -0.1128 0.2665 -0.0416 -1.3979 4.31 H

M-2 -1.5099 -0.1128 0.3548 -0.0508 -1.3161 1.63

M-1 -1.5099 -0.1128 0.2513 -0.0556 -1.4271 6.27 T

M-2 -1.5099 -0.1128 0.3593 -0.0508 -1.2733 5.04

M-1 -1.5099 -0.1128 0.2597 -0.0432 -1.4063 4.88 IU

M-2 -1.5099 -0.1128 0.3552 -0.0566 -1.2733 5.04

M-1 -1.5099 -0.1128 0.2573 -0.0444 -1.4063 4.88 F

M-2 -1.5099 -0.1128 0.3552 -0.0057 -1.2732 5.03

M-1 -1.5099 -0.1128 0.2644 -0.3726 -1.3956 4.15 S

M-2 -1.5099 -0.1128 0.3517 -0.0054 -1.2765 4.78

M-1 -1.5099 -0.1128 0.2657 -0.0455 -1.4058 4.85 HS

M-2 -1.5099 -0.1128 0.3565 -0.0487 -1.3150 1.70

-1.3376

-1.386,

-1.225,

-1.102


pressure for Strontium (Sr)


Model

(M) Ei E0 E1 E2

T.E.

(Ryd)

%

error Exp.

Others [7.64] [7.65]

M-1 -1.5949 -0.3174 0.4512 -0.0338 -1.4944 -- H

M-2 -1.5949 -0.3174 0.4167 -0.0883 -1.6012 --

M-1 -1.5949 -0.3174 0.4126 -0.0498 -1.5489 -- T

M-2 -1.5949 -0.3174 0.4173 -0.0678 -1.5834 --

M-1 -1.5949 -0.3174 0.4079 -0.0330 -1.5369 -- IU

M-2 -1.5949 -0.3174 0.4786 -0.0000 -1.4333 --

M-1 -1.5949 -0.3174 0.4022 -0.0352 -1.5448 -- F

M-2 -1.5949 -0.3174 0.4785 -0.0001 -1.4333 --

M-1 -1.5949 -0.3174 0.4328 -0.0214 -1.5003 -- S

M-2 -1.5949 -0.3174 0.4796 -0.0012 -1.4322 --

M-1 -1.5949 -0.3174 0.4389 -0.0385 -1.5119 -- HS

M-2 -1.5949 -0.3174 0.45190 -0.1368 -1.5967 --

-- -1.1621

-1.2770

CHAPTER-7 295



pressure for Barium (Ba).

Variation contribution of energy LFCF

Model


%

error Exp.

Others [7.64] [7.65]

M-1 -1.5325 -0.3214 0.4427 -0.0322 -1.4434 -- H

M-2 -1.5325 -0.3214 0.4349 -0.1342 -1.5531 --

M-1 -1.5325 -0.3214 0.4014 -0.0483 -1.5008 -- T

M-2 -1.5325 -0.3214 0.4282 -0.1413 -1.5671 --

M-1 -1.5325 -0.3214 0.3738 -0.0419 -1.5220 -- IU

M-2 -1.5325 -0.3214 0.4438 -0.0005 -1.4107 --

M-1 -1.5325 -0.3214 0.3627 -0.0476 -1.5388 -- F

M-2 -1.5325 -0.3214 0.4436 -0.0006 -1.4108 --

M-1 -1.5325 -0.3214 0.4122 -0.0244 -1.4642 -- S

M-2 -1.5325 -0.3214 0.4461 -0.0006 -1.4084 --

M-1 -1.5325 -0.3214 0.4297 -0.0365 -1.4607 -- HS

M-2 -1.5325 -0.3214 0.4331 -0.1349 -1.5558 --

-- -0.9717

-1.2181


pressure for NiAl.

Variation contribution of energy(Ryd)

LFCF Model

(M) Ei E0 E1 E2

T.E.

(Ryd)

%

error Exp. Others

M-1 -2.5541 -0.1118 0.4598 -0.0898 -2.2959 -- H

M-2 -2.5541 -0.1118 0.6361 -0.1187 -2.1485 --

M-1 -2.5541 -0.1118 0.4219 -0.1278 -2.3719 -- T

M-2 -2.5541 -0.1118 0.6419 -0.1254 -2.1495 --

M-1 -2.5541 -0.1118 0.4396 -0.0981 -2.3244 -- IU

M-2 -2.5541 -0.1118 0.6310 -0.0080 -2.0430 --

M-1 -2.5541 -0.1118 0.4355 -0.1004 -2.3309 -- F

M-2 -2.5541 -0.1118 0.6310 -0.0081 -2.0430 --

M-1 -2.5541 -0.1118 0.4546 -0.0796 -2.2909 -- S

M-2 -2.5541 -0.1118 0.6245 -0.0078 -2.0492 --

M-1 -2.5541 -0.1118 0.4491 -0.1010 -2.3178 -- HS

M-2 -2.5541 -0.1118 0.6388 -0.1202 -2.1474 --

-- --

CHAPTER-7 296



pressure for Ni3Al.


Model

(M) Ei E0 E1 E2

T.E.

(Ryd)

%

error Exp. Others

M-1 -1.4887 -1.2089 0.2687 -0.0398 -1.3808 -- H

M-2 -1.4887 -1.2089 0.3539 -0.0484 -1.3040 --

M-1 -1.4887 -1.2089 0.2534 -0.0420 -1.4096 -- T

M-2 -1.4887 -1.2089 0.3582 -0.0510 -1.3024 --

M-1 -1.4887 -1.2089 0.2615 -0.0415 -1.3933 -- IU

M-2 -1.4887 -1.2089 0.3546 -0.0050 -1.2602 --

M-1 -1.4887 -1.2089 0.2590 -0.0462 -1.3933 -- F

M-2 -1.4887 -1.2089 0.3546 -0.0051 -1.2689 --

M-1 -1.4887 -1.2089 0.2661 -0.0356 -1.3791 -- S

M-2 -1.4887 -1.2089 0.3514 -0.0491 -1.2631 --

M-1 -1.4887 -1.2089 0.2647 -0.0437 -1.3887 -- HS

M-2 -1.4887 -1.2089 0.3556 -0.0490 -1.3030 --

-- --


pressure for Al2Sr.

Variation contribution of energy(Ryd)

LFCF Model

(M) Ei E0 E1 E2

T.E.

(Ryd)

%

error Exp. Others

M-1 -4.1274 -0.2464 0.8776 -0.1518 -3.6479 -- H

M-2 -4.1274 -0.2464 1.1473 -0.3684 -3.5946 --

M-1 -4.1274 -0.2464 0.7746 -0.2380 -3.8371 -- T

M-2 -4.1274 -0.2464 1.1385 -0.3879 -3.6231 --

M-1 -4.1274 -0.2464 0.8083 -0.1677 -3.7331 -- IU

M-2 -4.1274 -0.2464 1.1304 -0.0033 -3.2467 --

M-1 -4.1274 -0.2464 0.8013 -0.1715 -3.7440 -- F

M-2 -4.1274 -0.2464 1.1305 -0.0033 -3.2466 --

M-1 -4.1274 -0.2464 0.8475 -0.1290 -3.6552 -- S

M-2 -4.1274 -0.2464 1.1245 -0.0032 -3.2525 --

M-1 -4.1274 -0.2464 0.8422 -0.1804 -3.7120 -- HS

M-2 -4.1274 -0.2464 1.1458 -0.3723 -3.6003 --

-- --

CHAPTER-7 297


Following points are inferred from the careful analysis of these results:

From Table 7.2 we can see that the total crystal energy for the pure Aluminum

matches well with the experimental values [7.62]. The total crystal energy as

calculated using Model-2 along with S function shows 1.23 % deviation from

experimental data. For both the model potentials the percentile deviation is

shown in the Table 7.2. It can be seen that the best results are obtained by

Model-2 in combination with the S local field correction functions [7.39, 7.63,

7.64].

From Table 7.4 it is observed that the total crystal energy for the Chromium

achieves a good agreement with the experimental values [7.67] and other

available results [7.15, 7.68]. In case of Model-1 along with H screening

function shows the minimum percentage deviation comes out than the other

screening functions, while for Model-2 and S functions shows 3.63 %

deviation with experimental data and matches well with the experimental

results among the other screening functions. From percentage deviations it is

found out the present results computed using the Model-1 and Model-2 along

with six different local field correction functions show the good agreement

with available experimental and other data [7.15, 7.67, 7.68].

From Table 7.5 and 7.6 we can see that the total crystal energy for the pure

Nickel and Copper matches well with the experimental values [7.67]

respectively. The total crystal energy as calculated using Model-2 along with

H function shows 1.71 % for Ni and 1.63% for Cu deviation from

experimental data and it is observed that minimum deviation than the other

screening functions. From Table 7.4 and 7.5 it observed that the present

results computed using the Model-1 and Model-2 show the better agreement

with the other available results, for both the model potentials the percentile

deviation is shown in Table 7.4 and 7.5 for Ni and Cu metal respectively. It

can be seen that the best results are obtained by Model-2 along with H local

field correction function.

Total crystal energy for Ca, Sr and Ba are shown in Table 7.3, 7.7 and 7.8

respectively. From table 7.3, 7.7 and 7.8 it is observed that the maximum total

crystal energy is obtained due to Model-1 potential along with the H screening

functions. In case of Model-2 we can see that the computed values of total

CHAPTER-7 298


crystal energy due to IU, F and S screening functions and due to H & HS are

nearly same. It is also observed that the present values are in good agreement

with other values [7.65, 7.66]. Due to lack of other information we do not put

any concert remark for the total energy of Ca, Sr and Ba, but in the absence of

other data these results will provide a good set of data for future comparison.

For the binary alloys NiAl [7.47], Ni3Al [7.47] and Al2Sr [7.48] show the FCC

phase. Here we have first time reported the total crystal energy for the three

different binary alloys. Table 7.9, 7.10 and 7.11 show the various contribution

of total crystal energy at zero pressure for NiAl, Ni3Al and Al2Sr respectively.

From Table 7.8, 7.9 and 7.10 it is observed that the maximum total crystal

energy obtained due to T functions for Model-1 and Model-2 potential

respectively while minimum total crystal energy obtained due to S and IU for

Model-1 and Model-2 potential respectively.

Out comes of the present results obtained due to Model-2 with IU, F and S

screening functions almost same values for all three binary alloys.

Over all it is observed that the total crystal energy computed using the Model-

2 potential is achieves good agreement with experimental and other results.

7.3.2 Pressure and Volume relation

The behavior of various materials under compression finds application in the

field of materials science and engineering technology. Hence the study of the

Pressure-Volume relations is important in condensed matter physics. The equation of

state for many crystals under pressure has been measured using various experimental

techniques. The data found by Bridgman (1948) [7.27] using the piston- cylinder

method were the first to be reported. However, the pressure scale at that time was

invariably old-fashioned so a correction for Bridgman’s compression data was

estimated. Up to the present time, data on the pressure-volume relations with other

experimental techniques such as shockwave and ultrasonic measurements have been

reported. A remarkably precise measurement of lattice compression using a high

pressure x-ray diffraction method has also been made by Senoo et al.[7.70] and

Grover et al.[7.63].

CHAPTER-7 299


Some theoretical calculations of the equations of state under pressure have

been reported for alkali metals using the pseudopotential formalism and the

perturbational theory. Also the pseudopotential formalism is applied to the

anharmonicity of lattice dynamics, the alloy system and the problem of complicated

lattice defects [7.39]. Sarkar and Sen [7.70] studied the equation of state using the

pseudopotential theory for Al. The equation of state for Al was calculated by Friedli

and Ashcroft [7.71] using the second order theory for the effective electron-ion

interaction.

In the present study we have calculated the pressure-volume relations for the

some alkaline earth metals (Al, Ca, Sr, and Ba) and some transition metals (Cr, Ni,

Cu) and solid solutions NiAl, Ni3Al [7.47], Al2Sr [7.48]. These solid binary alloys

have contain properties like light weight, high strength and hardness, low mass etc

and these alloys are particularly attractive for transport application such as

automobiles and weight reduction for industrial application. Al and Sr have been

found to be the two alkaline earth elements which affect high temperature

performance positively. Parallel to the experimental investigations, there is a great

need for an efficient route of alloy development with reduced cost and time. The

emerging concept of system materials design provides such opportunities.

Figure 7.1: P-V relation for Al using Model-1 & Model-2 potential along with Exp.

data[7.14] and Hugoniot data[7.13]

CHAPTER-7 300


Figure 7.2: P-V relation for Ca using Model-1 & Model-2 potential along with

Exp.[7.72]

Figure 7.3: P-V relation for Cr using Model-1 & Model-2 potential

Figure 7.4: P-V relation for Ni using Model-1 & Model-2 potential along with

other[7.15] and available Exp. data[7.44]

CHAPTER-7 301


Figure 7.5: P-V for Cu using Model-1 & Model-2 potential along with available other

data[7.15], Exp.data [7.44] and A. D. Chijioke et al. [7.13]

Figure 7.6: P-V relation for Sr using Model-1 & Model-2 potential along with

Exp.[7.72, 7.73], available other [7.74]and Vyas et al. data[7.75]

Figure 7.7: P-V relation for Ba using Model-1 & Model-2 potential along with

Exp. data [7.72]

CHAPTER-7 302


Figure 7.8: P-V relation for NiAl using Model-1 & Model-2 potential

Figure 7.9: P-V relation for Ni3Al using Model-1 & Model-2 potential

Figure 7.10: P-V relation for Al2Sr using Model-1 & Model-2 potential

CHAPTER-7 303


Figure 7.11: P-V relation for Al-Ni using Model-1 & Model-2 potential along with S

functions.

Figure 7.12: P-V relation for Al-Sr using Model-1 & Model-2 potential along with S

functions.

The pressure-volume (p-v) relation for pure Al has been shown in Figure 7.1.

It has been observed that the P-V relation calculated with T function shows

minimum deviation compared to other six local field correction functions for

Model-1 and it is observed that good agreement with the experimental results

[7.14] and Hungoniot data [7.13]. P-V relation computed using Model-2

potential gives the higher values than the experimental results [7.14] and IU

and F functions shows the maximum deviation compared to the other four

local field correction. Thus, Model-1 along with six local field correction

functions, the T function gives better results for P-V relation for pure Al.

CHAPTER-7 304


Figure 7.3 shows the P-V relation for Cr. Out comes of the present results of

Model-2 show higher numerical values compared to those computed due to

Model-1 along with all the six screening functions. P-V relation computed

using IU and F screening functions overlapping each other in both the model

potential.

From Figure 7.4 and 7.5 it is apparent that the P-V relation for Ni and Cu

along with the available experimental and other data [7.15, 7.44]. Out comes

of the present results for EOS are not found in good agreement with the

experimental and other data [7.15] because they might have taken the valency

higher than that employed in the present approach. It is observed in the region

(0.85 to 1.0) present results computed using Model-2 are found good

agreement with the other and experimental results [7.15, 7.44].

The computed P-V relation for Ca along H and S screening functions lye

along with experimental data [7.70] and hence this screening functions show

good agreement with experimental data than the other screening function

computed due to Model-1 potential. It is also observed that the present results

computed using Model-2 along with different screening functions is higher

than the available experimental data.

Figure 7.6, Shows the P-V relation for Sr compared along with experimental

data due to Budy and Strong [7.72, 7.73], other data reported by Vyas et

al.[7.74, 7.75]. It is observed that present results computed using the Model-1

and Model-2 along with Hartree (H) screening functions shows the better

agreement with the available of experimental data and obtained due to Vyas et

al.[7.75] results. To include the effect of temperature, Vyas et al. [7.75] have

reported on empirical EOS with the adjustable parameter and deduce EOS at

room temperature. Inflections in their EOS at pressures 24 and 63.5 Kbar

indicating structural phase transition cannot be reproduce by the present

scheme. On other hand, results on high-pressure [7.75] EOS predict SPT at

about 35Kbar a first-order transition from one crystal structure (fcc-cF4) into

another (bcc-cI2) [7.75] is observed.

Figure 7.7 shows the p-v relation for the Ba metal compared with available

experimental data [7.72]. At a first sight it is observed that the present results

computed using Model-1 potential are in excellent agreement with the

experimental results than the calculated using Model-2 potential. It is also

CHAPTER-7 305


observed that the outcome of present results calculated using Model-1 using

with T- function shows good agreement with experimental data computed

other screening functions. It is also observed that to IU and F screening

functions show that maximum deviation and H screening function shows that

minimum deviation with respect to the experimental results calculated as by

Model-2 potential.

Our present approach for some alkaline earth metals and some transitions

metals are found to be good agreement with available experimental and other

results. From this motivation we have extended our calculation part for the

binary system. As a case study for the first time we have calculated P-V

relation for NiAl, Ni3Al and Al2Sr binary system as shown in Figure 7.8, 7.9

and 7.10 respectively. In all the three binary systems, it has been observed that

the S function and IU as well as F functions show the maximum influence

with respect to H screening function for Model-1 and Model-2 respectively. P-

V relation computed using Model-2 potential is on the higher side than the

Model-1 in general.

From the Figure 7.11 and 7.12 it is seen that the equation of state for Al-Ni

and Al-Sr alloys lies in between the pure components.

7.3.3 Energy-volume relations

Energy-Volume Relations always serve as a database in the construction of

empherical schemes. Both self consistent and non-self consistent Harris Foulkes

calculation can be done and connection are made between them and simpler tight

binding and classical model of interatomic forces. Energy –Volume Relations help us

to find equilibrium lattice parameter and also to calculate elastic properties like bulk

modulus etc.

In the present study we have calculated the energy -volume relations for the

solid binary NiAl, Ni3Al and Al2Sr system with different concentrations of the

individual components.

CHAPTER-7 306


Figure 7.13: Energy and Volume relation for Al using Model-1 & Model-2 potential

Figure 7.14: Energy and Volume relation for Ca using Model-1 & Model-2 potential

Figure 7.15: Energy and Volume relation for Cr using Model-1 & Model-2 potential

CHAPTER-7 307


Figure 7.16: Energy and Volume relation for Ni using Model-1 & Model-2 potential

Figure 7.17: Energy and Volume relation for Cu using Model-1 & Model-2 potential

Figure 7.18: Energy and Volume relation for Sr using Model-1 & Model-2 potential

CHAPTER-7 308


Figure 7.19: Energy and Volume relation for Ba using Model-1 & Model-2 potential

Figure 7.20: Energy and Volume relation for NiAl using Model-1 & Model-2

potential

Figure 7.21: Energy and Volume relation for Ni3Al using Model-1 & Model-2

potential

CHAPTER-7 309


Figure 7.22: Energy and Volume relation for Al2Sr using Model-1 & Model-2

potential

For the Energy-Volume relation the following point is inferred form the above

figures:

Here we have compared the energy-volume relations of each metals and alloys

with six local field correction functions (LFCF).

From the figures 7.13 to 7.22 for Model-1 potential we can infer that all the

local field correction functions show the similar trend with Farid et al. and

Ichimaru et al. functions show the very much similar result than the other

functions.

In all pure FCC metals (Al, Ca, Ni, Cu and Sr) computed using Model-1, T

screening function shows the maximum and S screening function shows the

minimum influence with respect to H function than the other five local field

correction functions. F screening function shows the maximum influence with

respect to the H functions than the other five screening functions for BCC (Cr,

Ba) metals.

Model-1 and Model-2 both give the same trend in energy with change in

volume, but in general the numerical values of energy for a partial on value of

volume using Model-1 are on the higher side.

Figure 7.20 shows the energy-volume relation for the NiAl binary system. For

the energy-volume relation the computed percentile influence with respect to

H function are 3.66%-3.31%, 1.04%-1.28%, 1.64%-1.56%,0.28%-0.18% and

CHAPTER-7 310


1.02-.95% for T, IU, F, S and HS screening functions respectively using

Model-1 and for Model-2 the percentile influence with respect to H function

are 0.23%-0.04%, 8.43%-4.91%, 8.44%-4.91%, 8.07%-4.63% and 0.034%-

0.05% for T, IU, F, S and HS screening functions respectively.

Figure 7.21 shows the energy –volume relation for Ni3Al binary system. For

the energy-volume relation the computed maximum and minimum percentile

change with respect to H function are found to be 2.12%-2.15%, and 0.15%-

0.72% for T, and S screening functions respectively using Model-1 and for

Model-2 potential the computed maximum and minimum percentile influence

with respect to H function are found to be 5.81%-3.37% and 0.15%-0.077%

for F and HS screening functions respectively.

Figure 7.22 shows the energy-volume relation for Al2Sr binary system. For the

energy-volume relation the maximum and minimum percentile change with

respect to H function is found to be 3.47%-5.18% for T and 1.90%-1.88% for

HS screening functions using Model-1 potential. For Model-2 potential

maximum and minimum percentile change with respect to H screening

function is found to be 21.48%-9.69% for F and 1.01%-0.58% for HS

screening function respectively.

Following points are inferred from the careful conclusions of these results for total

energy, p-v relation and E-V relations:

Lastly we conclude that the total energy computed using Model-2 potential

along with Sarkar et al. screening function is found to be a better agreement

with experimental data and other available theoretical data.

Total energy for the transition metals (Cr, Ni and Cu) computed using the

Model-2 along with H and HS screening functions are found to agree well

with the experimental data and other available results.

Model-1 potential along with the T function is found to generate overall higher

total energy than other screening functions.

Pressure –volume relation found for the alkaline earth metal using Model-1

potential show good agreement with experimental data and other available

data.

CHAPTER-7 311


Pure FCC metals (Al, Ca, Ni, Cu and Sr) using Model-1 potential the T

screening function shows the maximum influence with respect to H function

than the other five local field correction functions. For BCC (Cr, Ba) metals

using Model-1 potential the F function shows the maximum influence with

respect to the H functions than the other five screening functions.

Overall equation of state (EOS) for all alkaline earth metals we achieve the

good agreement with respect to the experimental data and other data.

CHAPTER-7 312


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