Parabolic laws of the surrounded-atom model from ab initio calculations on clusters

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Parabolic laws of the surrounded-atom model from ab

initio calculations on clusters

A. Pellegatti, F. Marinelli, J.-C. Mathieu

To cite this version:

121

Parabolic

laws of

the

surrounded-atom

model

from ab initio

calculations

on

clusters

A.

_Pellegatti,

F. Marinelli and J.-C. Mathieu

Centre de

_{Thermodynamique}

et de Microcalorimétrie, UPR A7461 du CNRS, 26 rue du 141e R.I.A., F-13003 Marseille, France

(Reçu

le 14 avril 1989, révisé le 24

_juillet

1989,

accepté

le 15

septembre

1989)

Résumé. 2014 Dans le modèle de « l’atome entouré _», des

_hypothèses

_empiriques

concernant les

énergies

associées aux _supports élémentaires de

_{configuration}

dans les

_alliages

binaires, fonctions

paraboliques

de la concentration locale, ont été formulées. Dans cet _article, à

_partir

de calculs ab initio MO-CI sur des

_{petits agrégats,}

nous montrons _que ces lois

_paraboliques

ont un fondement au niveau

_{microscopique.}

Deux

_alliages

sont étudiés

_{[Li, Na]}

et

_{[Al, Li]}

et nous retrouvons

qualitativement

un ordre correct en ce

_qui

conceme la

_dissymétrie

de leurs courbes

_{d’enthalpies}

de

_mélange.

En fait, nos _{calculs montrent, dans les deux} cas étudiés, que deux lois

paraboliques

différentes sont _{nécessaires,} alors

_qu’une

seule loi avait été

_{prévue à l’origine.}

Abstract. 2014 The _« surrounded-atom » model formulates

empirical

hypotheses

on

_energies,

associated to

_{elementary configurational}

_supports of

binary alloys,

via

_parabolic

laws which are

functions of the local concentration. In this paper,

using

ab initio MO-CI calculations on clusters,

we have shown that these

_parabolic

laws have a

_microscopic

electronic basis. Two

_{binary alloys,}

[Li, Na]

and

_{[Al, Li],}

were studied and we

_{qualitatively}

found a correct order for the

_dissymetry

of their

_enthalpy

of

_mixing

curves. In fact, we found that, in both cases studied, two different

parabolic

laws were _{necessary, whereas}

_only

one _type of

_parabola

was

_{originally postulated.}

J.

_Phys.

France 51

₍₁₉₉₀₎

121-130 le’ JANVIER 1990,

Classification

Physics

Abstracts

05.70 - 31.20D - 65.50

1. Introduction.

The « surrounded atom »

(SA)

model was

_developed

at the end of the sixties

_by

Mathieu et al.

[1]

and

_{extensively applied}

to a wide range of metallic solutions

[2, 3]

to

_determine,

_among

other

_properties,

their

local,

or

_short-range,

order. It offered the

_advantage,

for

_example,

of

correctly describing unsymmetrical thermodynamical properties

of

_mixing

curves whereas the

quasichemical

model was

_only

successful for

_symmetrical

ones.

_Now,

more

_{sophisticated}

methods have been

_{developed by physicists}

such as the CVM

_(cluster

variation

_method)

_[4]

or the GPM

_(generalized

_perturbation

_{method) [5],}

and are

_currently

used in the theoretical

description

of

_alloys.

_However,

classical relations used

_{by metallurgists}

_(regular,

pseudo-regular

or

_sub-regular

_solutions,

_polynomial

_{representations)}

can

_naturally

be derived from the SA

_concepts

and formulation. In this sense, new

_insights

into the foundations of the model can still appear of

_topical

interest.

The SA method was

_{empirically parametrized}

in order to fit

_correctly

_{the widest range of}

binary alloy

properties.

In

_{particular, parabolic}

laws were derived to describe the variation of

the _{internal stabilization energy} as a function of the number of atoms B

_surrounding

the central atom A in SA entities

_{A (z + 1 - j) Bj (0 -- j -- z ),}

z

being

the average coordination

number involved in the

_{representation}

of the

_alloy

solution. The

_parabolic

form was found to

be that with the smallest number of free

_physical

_parameters

_(mainly,

_{partial enthalpies}

at

infinite

_dilutions)

and able to

_correctly

describe the

_shape

of the curves of various

_enthalpy

of

mixing.

The

_parabolas

_possess a horizontal

_tangent

_{for j}

= z.

As will be seen in section

_2.1,

one of the

_assumptions

on which the SA method is based is that the solution is infinite. Solid state

_physics

calculations have shown that such

_parabolic

laws are valid in many cases

[6]

but not for all

_types

of

_{binary alloys,}

even when limited to the first transition metal series

_[7].

In these calculations the

_infinity

of the solution is

_properly

taken into account but the electronic _{energy evaluation is carried} out

_only

at a

_{semi-empirical}

level.

In _{this paper} we

_approach

the

_problem

from another

_point

of view. In

fact,

we _{may wonder} whether these laws have a real

_significance

at the

_microscopic

level of the cluster or if

_they

are

only

a _{consequence of both the local electronic} structure and the

_infinity

of the solution. In order to show that

_parabolic

laws are connected with

_{microscopic properties}

ab initio

quantum

chemistry

methods can be

_adopted

to obtain reliable cluster

_energies.

In the next section we

_briefly

_report

on the SA

_method,

_emphasizing

the main

hypotheses

and formulas which lead to the choice of the

_energetic

laws. Section 3 is devoted to the ab initio determination of the cluster

_energies.

Our choices of cluster size and

type

of

_binary

alloys

will be discussed in order to show that at least the

_following

two conditions must be fulfilled :

_i)

the clusters must be

_{representative}

of a real

_alloy

_solution,

_ii)

the atoms involved

must be characterized

_{by significantly}

different

_enthalpy

of

_mixing.

The

_adopted

ab initio molecular orbital

_(MO)

method with

_{configuration}

interaction

_(CI)

will be

_{presented. Finally,}

from the stabilization

_energies

of the

_clusters,

the

_parameters

of the fitted

_{representative}

curves are derived. Our results will be discussed both from a

_quantum

chemical

_point

of view and on the basis of

_{thermodynamical}

considerations.

2. The surrounded-atom model.

2.1 BASIC RELATIONS. - The

_hypotheses

of the

_regular

solution model

_provide

the basis of the SA model

_[1].

_They

include : _{the total energy}

_developed

in a sum of contributions from each

_{elementary configurational}

_support,

no

_long-range

interactions taken in account, and the

quasi-lattice

structure of the

_liquid.

The main characteristic of the SA model is the

_elementary

configurational

support

as a central atom in the force field of its z nearest

_neighbours

instead of the usual

_pair

interactions,

z

_{being independent}

of the concentration. Table 1 shows the various contributions to the

_potential

_{energy of the solution from each SA}

_type.

In this

table,

for a central atom of

_type

_A,

_Ci

are binomial

coefficients,

N the total number of atoms,

mi

the

probability

of

finding

one atom A with the

_surrounding

_{A(z-j) Bj}

in the

solution,

EjA

its energy, and the

corresponding

formulas

_(i

instead

_{of j}

and

_p’s

_{replacing m’s)}

for SA’s with a central atom of

_type

B.

123

Table 1. - Contributions to the

_potential

_energy

_of

the solution

_from

each SA

_type.

lead to the

_following

relation

If the

_experimental

conditions are such that the

_enthalpy

can be identified with the internal energy, the

enthalpy

of

mixing

is derived from relation

(1)

after a statistical treatment

where

_Uj

and

_Vi

i are the surrounded atom differences

and

_mi

and

_pi

are mean values of

_{probabilities}

which will be obtained from

_subsequent

statistical treatments.

_Indeed,

statistical treatments

_simplify

the

_partition

function

_using

the

potential

energy

(1),

replacing

the

complexity

of the solution

by

a « mean

_{configuration}

».

Independently

of the statistical treatment, we had to face the

_problem

of

_determining

all

Ujs

and

_{Vi’s. Originally, they}

were

_empirically

determined. In

_general,

this means that 2 z

_parameters

would have been necessary to

_get

an

_enthalpy

of

_mixing

curve for a

_{given alloy.}

In

fact,

Mathieu showed that

_{two-parameter}

laws of variation of

_Uj

and

_Vi

_{(one-parameter}

laws in very

simple

but rare cases of almost

_symmetrical

curves of

_enthalpy

of

_mixing)

were sufficient to describe a _{wide range of}

_enthalpy

of

_mixing

curves

_correctly.

The stated

_parabolic

laws had the form :

interpret-ation)

introduces an

_equilibrium.

For

_example

in the case of a central atom A

AE = 0

corresponds

to a linear law of variation while AE = Cst.

corresponds

_{to a}

parabolic

law of variation.

Then,

statistical methods at various levels of

_{sophistication}

_(complete

_disorder,

quasichem-ical treatment,

...)

were used to calculate the numbers of SA of each

_type

via the evaluation of

probabilities mj

and pi.

In the

_{simple Bragg-Williams}

statistics,

analytic

formulas are

_easily

derived,

whereas in the

_Guggenheim

treatment, solutions are found

_{only through}

the

resolution of a secular

_equation.

2.2 HYPOTHESES ON SURROUNDED-ATOM ENTITIES. - In the

original,

and more

_widely

used,

SA

method,

a first

_hypothesis

states

_that,

in an SA

_{entity, peripheric}

atoms of a

_given

type

are

_{indistinguishable. Peripheric-peripheric}

atomic interactions are not taken into

account. This also means that

_{regular polyhedra}

are the best choice to

_represent

the SA entities. On the other

_hand,

no

_explicit

information is available as for the

_shape

of these

polyhedra

and more

_precisely

about the interatomic distances. In a real lattice in the solid

state these distances

_might

be all

_{equal, independently}

of the nature of atoms A and B. In the

liquid

state, the situation is less

clear,

since the average coordination number z

is,

in this _case,

considered as a

_parameter

with non-conventional values. In

_liquid,

distortions could turn out to _{be necessary in order} to close the

_{quasi-lattice.}

We shall come back to these

_geometrical

considerations when

_discussing

our choice of model clusters in the next section. 3.

Quantum

chemical

_approach

to surrounded atoms.

We consider SA

_{configuration}

entities as isolated clusters. The

_U/s

and

_Vi’s

are small

quantities

in

_comparison

with total

_energies

of

_clusters,

since

_they

are defined as differences of stabilization

_energies.

Reliable values for such small

_quantities

can be obtained

_only

by ab

initio

_quantum

chemical

_methods,

_using

extended basis sets and

_algorithms

able to

_give

a reasonable amount of

_electronic

_{correlation energy.}

3.1 CHOICE OF MODEL CLUSTERS. In this section we must choose first the structure of the clusters and second the

_type

of

_{binary alloy}

we treat. In connection with the first

_point

it must

be clear that we shall not be

_{automatically}

interested in

_determining

_{the minimum energy}

ground-state

of each cluster

_(which

in

_general

is the first task of a

_quantum

chemist).

Indeed,

since the energy laws of the SA model that we would like to

_verify

have been established and used under well-defined

conditions,

we have to first

_try

to fulfil these conditions in our calculations on clusters. The main conclusion we can draw from section 2.2 is that the

polyhedra

used in the SA model are «

hypothetically-averaged

» ones. We shall not be able to

reproduce exactly

all the

_hypotheses

of the SA model

_but,

in some cases, we shall

_try

to have a mean

representation

of them. For

example,

let us consider a

real liquid

binary

solution

AB,

where x is the molar fraction in B. Close to x =

0,

_we almost have the pure substance A and

rAA interatomic distances

prevail

throughout.

The reverse holds for x =

1,

where we have

rBB distances. Around x = 0.5

(in

the case of

no-demixing)

we

_certainly

have a

_great

amount of rAB distances in the solution.

_Averaged

distances can be considered as connected with the molar fraction. In the solid

_phase,

such

distortions,

even if

_they

exist,

are attenuated. The SA model treats both

_liquid

and solid

_phases

and we must choose a

_compromise.

We shall use

_regular

_polyhedra,

even

_though

ab initio

_{calculations,}

needed from an _energy

125

a coordination number z = 4 is

commonly

found in materials. Therefore we will use, as model

clusters,

centered

_regular

tetrahedra of five atoms with

_equal

distances.

_Energy

minimizations

of ten clusters have to be run in this way for one

_alloy.

We _{shall compare} two

_alloys,

one with an almost

_{symmetrical enthalpy}

of

_mixing

curve and the second with an

_{unsymmetrical}

one. From a numerical

_point

of

view,

we will

_try

to deal with the

_lightest

atoms

_possible.

The

[Na, Li]

and

_{[Al, Li] binary}

_alloys

can be considered

_good

candidates for our calculations. The aim of the SA method is

_mainly

to treat

_{liquid phases}

of

_{binary alloys}

_and,

at this level of

approximation, only averaged

coordination numbers are introduced. The coordination

number z = 4 is

certainly

too small to

_represent

the true structure of a metallic solution.

However,

at this

_point

we will not introduce

_temperature

effects and our

_goal

is not to

reproduce enthalpies

of

_{mixing quantitatively.}

In that sense, the choice of a tetrahedral

structure for our clusters cannot be considered a real drawback for these model cluster

calculations. Such

_elementary

five-atom cells which are

_commonly

found in other

solids,

can be considered as

_having

the minimum size to be models for a condensed

_phase.

The

approximation

is no more than some others in the method.

3.2 MO-CI CALCULATIONS. - The

_{pseudopotentials}

of Durand and Barthelat

_[8]

were used

to restrict the electronic

_problem

to valence electrons

_only.

At the SCF level we made use of the PSHONDO _program

_[9],

an extension of HONDO

_[10],

which includes

_{pseudopotentials.}

From extended atomic Gaussian basis sets

_[11],

we extracted a

_triple-es,

one

_single-£p

(contracted [3, 1, 1/2])

basis set for Li and Na atoms and a

_{double- £s}

_{and p}

_{(contracted [3, 1/3,}

1])

basis set for Al atoms. In calculations on similar clusters of these atoms, d-orbitals were found to have a

_{non-negligible}

influence in the determination of an _{energy minimum}

_[12].

But the

_opposite

conclusion was reached

_by

other authors

_[13].

_Being

_{interested in energy}

differences,

we did not introduce d-orbital contribution in our calculations.

_However,

p-orbitals of Al atoms were

_represented

at a

_double-C

level.

Electronic correlation is introduced

_by

means of CIPSI

_algorithm

_[14].

In the first

_part

of this

_method,

a variational CI is carried out in a

_subspace

built from the most

_important

determinants

_(determined,

for

_example

at the first

iteration,

from the order of MO

_levels)

for all the states studied. This

_gives

a

_{multiconfigurational}

zeroth-order wave function

which,

in a second

_step,

is

_perturbed

_up to _{second order in energy. New}

_{configurations,}

whose

_weights

in the

_{perturbation expansion}

are found to be

_higher

than a

_given

_threshold,

are included in the variational treatment. _{Then the process is}

_repeated

_{until the desired convergency in the}

perturbation

energy is reached. For the

perturbation,

the Hamiltonian was

_partitioned

according

to the

_{Epstein-Nesbet}

definition

_[15].

To end this section three technical remarks must be added.

_First,

for alkali atom

_clusters,

core-valence correlation

_energies

_{may have} a

_{non-negligible}

influence. This influence was, for

example, emphasized by Jeung

et al. who

_proposed

a treatment based on the fluctuation of the electric field

_by

_{perturbation theory}

_[16].

More

_recently,

Müller et al.

_proposed

the use of

core

_{polarization potentials}

_(CPP)

with

_only

a

_{single adjustable}

atomic

_parameter

_[17].

_High

accuracy is then obtained for the

ground-state spectroscopic

constants of alkali dimers and mixed alcaline diatomic molecules. In

_fact,

_{many authors in the}

_past

_brought

their own

contributions to the solution of the

_problem

and an exhaustive discussion can be found in the papers of Müller et al.

[17].

Though meaningless

to our _{purpose in this paper,}

_Uj’s

and

V,’s energies

for these diatomic clusters can be evaluated from

_De

values in table IV of the second reference in

_[17].

We have to _compare

_{CI(v) (valence CI), CI(v)}

+ CPP and

experimental

results. For a « central » atom

_Li,

the molecule

_Li2

is the

reference,

and

Ui

values for LiNa are : +

0.178,

+ 0.176 and + 0.180 eV

_{respectively.}

For a « central » atom

closer to the

_experimental

one than that

_{corresponding}

to the

_CI(v)

_{approximation. Though}

the differences are _very

_weak,

this is not the case for

_Ui

values. Even for extended basis calculations on dimers

_(certainly

due to error

_{compensations) CI(v)}

_results,

_{for the energy} differences we are interested

_in,

can be closer to the

_experimental

ones than those which are obtained from the more

_{performing CI(v)}

+ CPP

_{approximation.}

With clusters of five atoms,

we do not

_employ

such

_large

basis sets and the addition of

_CPP’s,

or

_{equivalent procedures,}

may appear like refinements with

finally

no absolute certitude of

_improving

_Uj

and

Vi energies.

Therefore,

we

_neglect

core-valence correlation

_{energies. Secondly,}

for a

_given

cluster,

the same variational

_subspace

is used around the minimum on the

_potential

_energy curve.

_Therefore,

the

_subspace

at each

_step

must include all the determinants

_significant

in the range of interatomic _{distances for which the energy} curve is calculated.

_Finally,

in order to

obtain a

_{meaningful comparison}

between

clusters,

the

_perturbative

_part

of the correlation energy must be obtained in a _{uniform way. In}

fact,

for all clusters the criterion chosen was that the sizes of variational and

_{perturbational}

contributions to the correlation _energy are

equivalent.

The energy of the five isolated atoms has been evaluated under the same conditions.

Table II.

_{- Energies of}

_{[Na, Li]}

clusters in a tetrahedral structure. r is the

_equilibrium

bond

length

between nearest

_{neighbours ;}

E is _{the total energy ;}

_Ed

is the

_complete

dissociation

energy. The type

of atom

located at the centre

_{o f the}

tetrahedron is indicated

_first.

All values are in a. u.

Table III. -

127

3.3 CLUSTERS ENERGIES AND SA-PARAMETERS DERIVATION. - In tables II and

III,

we

report

the

_symmetry

of the lowest state in the tetrahedral

_geometry,

the

_equilibrium

nearest

neighbour

distance,

the total energy, the dissociation energy and the difference of the

dissociation

_energies

between the

_{j-substituted}

_{(resp. i)}

and the non-substituted cluster. The

Ui

and

_Vi

differences are

_plotted

in

_figures

1 and 2 which show

_clearly

that

_parabolic

_{laws very}

correctly

describe the variations of

_Ui

and

_Vi

as a function

_{of j}

_{(resp. i)}

for both

_alloys.

Two

types

of

parabolas

have been found :

_aj2

and -

_{8 i}

₍₂

z -

i ).

Fitted values for a

and P

are

given

in table IV.

Fig. 1.

Fig.

2.

Fig.

1. -

Shape

of the curves

_Uj

=

f(j)

and

Vi

= g (i )

for the

_{[Na, Li]}

_binary

_system. Crosses

correspond

to exact values of cluster energy differences. Curves

correspond

to _average

_parabolas

of the type :

a j2 or - f3 j (8 - j)

(resp. i).

The type of the central atom is indicated.

Fig.

2. -

Shape

of the

_{curves Uj}

=

f(j)

and

Vi

=

g(i)

for the

[Al, Li]

binary

sytem. See

figure

1

caption

for

_{explanations.}

Table IV. - a and

_f3

constants

_of

the

_parabolic

laws in the

_SA-model,

extracted

_from

the

4. Discussion.

4.1

QUANTUM

CHEMICAL CONSIDERATIONS.

4.1.1 Electronic structure

_of

the clusters. - Five-atom alkali metal clusters

are

_generally

Jahn-Teller unstable in a tetrahedral structure. In

fact,

the calculations

_reported

in the literature

mainly

concem other structures which are more suitable to describe the

_ground

state.

Alkaline clusters have been the most studied

_species

and a review was

_{given by Koutecky}

and

Fantucci

_[18].

For

_Li5

and

_Nas

_{lowest energy} structures are a _square

_pyramid,

a

_triangular

bipyramid

or a

_planar

set of three

_triangles

_[18].

Also accurate results on

_Als

can be found in the recent literature. In

_general

the

_reported

ground

state or state close to the

_ground

one structures are those mentioned above for alkaline clusters. However the real electronic structure of the

_ground

state is still not

completely

clear. Some calculations

_suggest

a

_quartet

state in a _square

_pyramid

structure

_[13]

whereas other calculations

_give

a doublet state for the

_{planar C2v-type}

structure

_[19].

Mixed five-atom clusters of such elements are not found in the literature.

In the pure alkaline series all clusters are in a

_quartet

state. In the aluminium

series,

only

two clusters are in a

_quartet

state and all the others in a doublet state. But the

_geometric

structure is not the same and we cannot _{go ahead for direct}

_comparison

with other calculations.

_Indeed,

as

_emphasized

in section

_3.1,

the interest of this paper lies not in

considering geometric

structures

_{corresponding}

to

_ground

_states, but rather in

_considering

a

structure common to all clusters and

_{energetically}

unstable as

_they

_may

_be,

in order to

approach

some 3D-structure

_type

and the SA

_hypotheses.

Let us remark that the bond

_lengths

calculated in the

_present

_study

are of the same order of

_magnitude

as those obtained

_by

other

authors,

for

_example

for

_Als

_[19].

One recent work treats the case of

_BeLik

_(k

= 1 -

9)

clusters

_[20].

For k =

4,

the

ground

state

(lAlg)

is of

D4h

_symmetry.

A

higher multiplicity

state

_{(3T1 )}

is found in the case of a

_Td

_symmetry.

The basis set size was found to have

_only

a

secondary importance

at least for the

_qualitative

trends of cluster

_properties

found in their work. These results are not in contradiction with ours. No

_{systematic study}

has

_yet

been made for mixed-clusters of five atoms.

4.1.2 Parabolic laws

_of

the SA-method. -

Only

one

_type

of

_parabolas

(relations (4))

with

empirically

fitted

_parameters

were _necessary to the first users of the SA method for

_correctly

describing enthalpies

of

_mixing

of

_{binary alloys.}

In the

_present

_{investigation,}

two

_types

of

parabolas

are found to be

_important.

As a _{consequence, the}

_energetics

_accompanying

the first atomic substitutions in the

_peripheric

shell in

_comparison

with those associated with

subsequent

substitutions

_depend

on the

_type

of atoms

_constituting

the

_{binary alloy.}

Of course, due to the limited number of the cases

_studied,

we cannot

_try

to

_give

a definitive

explanation

of the variation laws.

_Only

the

_following

comments can be formulated :

i)

when the

_substituting

atoms are more

_{electronegative}

than the central atom, the absolute value of the stabilization _{energy increases with the number of}

_substituting

atoms and vice versa. A direct consequence is the

_sign

of the constant in the

_parabola

formula and therefore

the direction of its curvature.

_However,

there is no direct influence on the

_type

of the

parabola,

which is different for

_example

in the series

_[Na.,

_Liy]

and

_[Li,,

_Aly]

on the one hand and

_[Lix

_{Nay ]}

and

_[Alx

_Liy]

on the

_{other ;}

ii)

these laws do not seem to be

_directly

connected with the

_substituting

atomic size. From

tables of bond

_lengths

_[21],

we can evaluate atomic sizes.

_Here,

in

_decreasing

order we

roughly

have : Al Li Na. It is reasonable to think that it would be more and more difficult

to make substitutions

_by

atoms

_larger

than those of the

_original

_pure

_{compound -}

which

129

[Na, Li]

system.

In the latter _case, we

_only

observe either dilatation of

shrinking

of bonds as we substitute

_larger

or smaller atoms

_respectively

_{from the pure}

_compound

_{representative}

cluster.

Very

accurate results are available for triatomic mixed alkaline clusters

_[22].

Of course, the number of atoms is too small to

_represent

even the

_beginning

of any

crystalline

lattice.

Moreover,

their structures

_{being fully optimized, they}

remain far from the

_{SA-hypotheses.}

However,

after evaluation of

_{U i 1 s}

and

_Vi’s,

we

_already

observe a

_tendency

toward

parabolic

laws in the case of these very small clusters.

4.2 THERMODYNAMICAL CONSIDERATIONS. - The

_energies

_Uj

and

_Vi

have been determined

ab initio. Relation

₍₂₎

is the basic relation from which to

_get

the

_enthalpy

of

_{mixing, provided}

we have a _{way of}

_evaluating

the average

_{probabilities mi}

_{and Pi}

for each

alloy.

This is the work of statistical methods.

Here,

we would like

_only

to _{consider - and compare for the} two

_alloys

studied - the

_{dissymetrical}

character of the

_enthalpy

of

_mixing

curve with

respect

to the molar fraction x = 0.5. This can be done

by comparing « partial enthalpies

of

mixing

at

infinite dilution ».

They

are

_{given by}

[1] :

The ratio p =

âHf / âH:

is a measure of the

_dissymmetry.

From the values

_given

in table IV

we obtain :

and

In these cases, the

greatest

partial enthalpy

of

_mixing

at infinite dilution

_corresponds

to the almost pure substance of the less

electronegative

atoms of the

_{binary alloy.}

The

_dissymetry

of the

_enthalpy

of

_mixing

curve with

respect

to the half

concentration,

greater

- but not too

much - for the

_{[Al, Li]}

_alloy

than for the

_{[Li, Na]}

_alloy,

is well found since the lowest value of _p

_corrresponds

to the

_{[Al, Li]}

_alloy.

_Moreover,

the results

₍₇₎

show that the

_enthalpy

of

mixing

for the

_{[Na, Li]}

_alloy

is

_positive

while for the

_{[Al, Li]}

_alloy

it is

_{negative. Only}

a few

experiments

have been made on these

_systems.

But we know that for the alkaline

_binary

alloys

in

_general,

the

_enthalpy

of

_mixing

is

_{slightly positive}

and that for the

_{[Al, Li]}

_alloy

it is

negative

since,

at x =

0.5,

_we observe the formation of a definite

compound.

5. Conclusions.

From these calculations on clusters we _{have proven that the}

_parabolic

laws of the

SA-method,

empirically

found in the context of a statistical

method,

have an electronic

_microscopic

basis. But instead of a

_unique

type

of

_parabola,

we

_found,

from our

_{calculations,}

two

_types

of

parabolas

which

_depend

on the central and

_peripheric

atoms

_present

in the series. The first

type,

which is the same as the

_original

one, has a nul derivative

for j

= z - the

_completely

substituted cluster - and the second

type

has a nul derivative

_{for j}

=

0,

i.e. the pure

substance.

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₍₁₉₆₅₎

1297.

[2]

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