LIQUID TRANSITION METALS AND ALLOYS

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LIQUID TRANSITION METALS AND ALLOYS

J. Gaspard

To cite this version:

J. Gaspard. LIQUID TRANSITION METALS AND ALLOYS. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-127-C4-130. �10.1051/jphyscol:1974422�. �jpa-00215613�

JOURNAL DE PHYSIQUE Colloque C4, suppliment au no 5, Tome 35, Mai 1974, page C4-127

LIQUID TRANSITION METALS AND ALLOYS

J. P. GASPARD

Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex, France and UniversitC de Likge, Sart-Tilman, 4000 Liege, Belgique

R6sum6. - Pour des systemes dkcrits par un hamiltonien de Iiaisons fortes, la methode des moments et le developpement de la fonction de Green en fractions continues permettent d'obtenir des densites dyetats corrects, meme avec un nombre limite de moments. De plus, la methode des moments est applicable ti des systkmes d'atomes simules sur ordinateur, ce qui permet d'ktudier une grande variete de types de dbordres. Nous avons applique ces techniques au cas des metaux de transition Iiquides et de leurs alliages, pour Iesquels nous avons montrh qu'une description prkise de I'ordre local est requise.

Abstract. - For systems described by a tight binding hamiltonian, the moments' method and the Green function expansion into a continued fraction allow for correct densities of states, even when a small number of moments is known. Moreover, the moments' method is applicable to computer simulated samples what allows for the study of a large variety of disorders. We have applied these techniques to the case of liquid transition metals and alloys. We have shown that a correct description of the local order is required.

1. Introduction. - In the tight binding approxi- mations, the calculation of the average Green function for topologically disordered systems presents two major difficulties.

First, little is known about the local order : X-ray or neutron experiments give a fairly accurate pair correlation function g,(r), but the n-body (n 2 3) correlation functions cannot be obtained experimen- tally. The Kirkwood superposition approximation is not completely satisfactory for the n-body (n 3) correlation functions, and moreover it is impossible to include it in analytical theories : only a weak version of the Kirkwood approximation can lead to tractable calculations ; for example g,(r) is taken into account only between atoms along a chain. In the cases where the short range order is not very important like for an impurity band, these further assumptions are justified.

On the contrary, when the packing fraction is large like for liquids, a careful description of the local order is required.

Secondly, it is impossible to average exactly the Green function, i. e. each term of the multiple scatter- ing expansion can be separately computed and averag- ed, but the summation of all terms is out of question.

For completely disordered systems, different approxi- mations were proposed in which all single site terms are summed up and the other terms are either neglected [I] or approximated [2]. If a short range order occurs, the concept of single site is less useful due to the fact that all atoms are linked to each other. However, similar approximations were developed. For most of them, the overlap integrals P(r) are replaced by P(r) g,(r) (extended chain approximation) [3, 4, 51.

Our approach is completely different and comple- mentary to these analytical mean field approximations.

It provides a numerical method which allows us to get rid of the two above stated difficulties. We generate on a computer a large cluster of atoms interacting through a potential chosen so as to fit the experimental pair correlation function. We compute the moments of the density of states for the central atom and we average it over all possible configurations of the cluster.

The main difficulty of this method is the finiteness of the cluster or equivalently the lack of an external selfconsistent field. We will show that the expansion in continued fractions removes partly this difficulty.

The simulation technique allows us to treat a very large variety of problems including liquid transition metals, liquid alloys, the effect of clustering or anti- clustering ... As an illustration of the possibilities of the method, we will show some results for different types of topological disorder.

2. Moments of the density of states. - As deve- loped in Cyrot's paper (this conference) [6] the moments of the tight-binding hamiltonian

are written

where yij is either pij or ei aij.

In the case of a topologically disordered system, like a pure liquid, the pij overlap integrals are random ; in the case of an alloy-type disorder, the diagonal

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974422

C4-128 3. P. GASPARD

elements E~ are random, while in the case of a liquid alloy both elements are random. One has to deal with quantities averaged over all possible configurations of the systems according to their constraints. To average analytically formula (2) is very difficult for the two reasons stated in the introduction. Let us look for instance at the fourth order moment :

(a) (b) ( 4 ( 4

single site terms pair term

In completely disordered systems, the terms (a), (b) and (c) are single site terms because they correspond either to self-avoiding terms (a) or can be exactly decoupled into such terms (b), (c). These ones can be exactly summed up [I]. The pair terms (d) can only be neglected [I] or approximated 121. When a short range order exists, the evaluation of integrals like (a) is very hard to perform and the difficulty becomes insolvable when the order of the moment increases.

A possibility for solving the problem is to assume that

(extended chain approximation).

However, in order to avoid a description of the local order through the n-body correlation function, one can apply formula (2) to the moments' computation of a cluster of atoms generated on a computer. The atoms interact through a potential V ( r ) which is deduc- ed from the pair correlation function g ( 2 ) (r). For simplicity we used a hard core potential which seems not too bad for describing the local order of liquid Ni [7].

The moments computed on the cluster take into account exactly the single site terms, as well as the pair, triplet ... terms involving the atoms of the cluster.

A difficulty of the method [is the statistical fluc- tuations, but for systems with a strong local order, like the liquid or amorphous substances, it is sufficient to average over a fairly small number of configurations (several tens). The second difficulty arising in compu- ter simulation is how to get rid of the size effects of the cluster or equivalently how to dress the cluster with a coherent external field. We will see in the next section that this problem is solved together with the conti- nuation of the continued fraction.

3. Density of states. - It has been proved [8]

that the usual methods of getting the density of states from is first moments (Legendre polynomials, Edger- worth series ...) are unable to give a correct density of states : they give spurious oscillations and mainly in the split band limit case, the density of states can become negative. The continued fraction expansion gives a positive defined density of states. Starting from the 2 n first moments, one expands the resolvant

into a continued fraction with n levels, o referring to the central atom of the cluster.

The coefficients a, (resp. b,) are functions of the 2 p - 1 (resp. 2 p ) first moments. In opposition to the other fitting methods, the coefficients a,, b, are very simply related to the band location i. e. for a single band, without any singularity due to long range order, like for liquid metals, the a, and b, are converging respectively towards the abscissa of the middle m of the band and the square of the bandwidth w divided by sixteen

For a band splitted into two subbands (alloys in the split band limit), these relations generalize easily : the a, and b, converge towards (or oscillate between) two limits related to the total bandwidth W and the gapwidth A by

where a , and b , are respectively the two limits of the coefficients a, and b,. One sees that the asymptotic values of the continued fraction coefficient depend only on the band location, whereas the way they

LIQUID TRANSITION METALS AND ALLOYS C4-129

converge depends on the band edges or internal singularities. It is interesting to notice that the coeffi- cients are converging much faster for disordered materials than for ordered ones. These theorems allow us to continue the coefficients of the continued fraction in accordance with exact results on the band location.

The continuation procedure has to be thought as assuming a coherent external field outside the clus- ter [9].

4. Application to liquid metals. - We compare the numerical results obtained by our method with two different analytical descriptions of liquid transition metals using the same hamiltonian for a non degenerate d-band. The overlap integral P(Rij) is assumed to decrease exponentially for large values of R i j : it has the form

B(R,) = - V, (I

+ g)

(

2)

where RB is the Bohr radius and V , is a scaling factor for the energy which is fitted to give the right band- width value. For liquid Ni at 1 500 K, RB is equal to 0.3 A while the hard core separation is about 2.2 &

and the packing fraction y is equal to 0.45. For these values of the parameters, the P(Rij) overlap integrals have a significant value only between nearest neigh- bours. Samples of 70 atoms were built, for which the 8 first moments of the most central atom were comput- ed in order to avoid surface effects. The averaging was performed over 30 samples. The density of states is plotted on figure 1 and compared to the results of

they remind of the density of states of a crystalline compact lattice (F. C. C. or H. C. P.) which looks natural due to the fact that the local order in liquid Ni is not very different of a compact packing of hard spheres, the average number of nearest neighbours being 11.7 [7].

5. Liquid alloys. Clustering and anticlustering effects. - We want to show that the simulation tech- nique is able to treat a very wide range of problems.

It seems to be specially interesting in very complicated cases where the analytical approaches fail to give an answer. As an illustration, we show on figure 2 the

I

ca = CB = 0.5, and a = 0

i j j

[ ⁽ full line), cr = 0.25 (broken line), cc = - 0.25 (dotted line).

50 1 ;

1 /

I ,,' \\\,

,,y ^{; \}

,,,+' / i ^'\ density of states obtained for a liquid alloy of transition

-.,,-,,,'

/.-'y,x i \\, metals far apart in the periodic table, i. e. in the split

-

*-: -

-

-

0 --A- :

band limit. The liquid is again simulated by hard

-002 - 0 0 1 o 001 Oo2 spheres.

The charge transfers are neglected : the atomic FIG. 1. -Density of states for a liquidrnetal(ful1 line). Thevalue level an is either &A Or &B

of the parameters correspond to liquid Ni at 1 500 K. Compa- the surrounding atoms, and PA,@) ⁼ P A B ( y ) = P B B ( ~ ) .

rison with Ishida's and Yonezawa's results (broken line) and One can also easily study the effects on the density

Roth's results (dotted line). of states of the clustering of atoms of the same nature (A or B). For completely random alloys, the probabi- lity yAB for an A atom to be the nearest neighbour of Ishida and Yonezawa [4] and L. Roth [IO]. The overall a atom is

bandwidth is quite similar in the three cases. Our

results show a broad peak in the positive energy region pAB = pBB = cB and similarly

and a plateau behaviour in the middle of the band : PAA ⁼ PBA ^{= C A} . ⁽¹⁰⁾

C4-130 J. P. GASPARD

The tendency for atoms of the same nature to cluster can be related to the a parameter defined by

PAA = (1 - a) cA

+

PBB = (1 - 01) CB $. = 1 - PBA . ^{(1 1)}

The random case is characterized by a = 0, the limit of complete separation by a = l and the complete anticlustering by a = - 1 and cA = 0.5. We see on figure 2 that for negative a values the band is much more peaked in its center, while it broadens for positive a values.

Conclusion. - The aim of this paper has been twofold. First, we wanted to emphasize that by the continued fraction method it is possible to get accurate density of states with a limited number of moments if the fraction is carefully continued. Secondly, we showed that the computer simulation technique is a complementary technique to the analytical approach of disordered systems. It provides useful infor- mations mainly in cases of complicated types of disorder.

References

[I] MATSUBARA, T. and TOYOZAWA, Y., Progr. Theor. Phys.

26 (1961) 739.

[2] CYROT-LACKMANN, F. and GASPARD, J. P., submitted to J. Phys. C.

[3] POPIELAWSKI, J., J. Chem. Phys. 53 (1970) 957.

[4] ISHIDA, Y. and YONEZAWA, F., Progr. Theor. Phys. 49 (1973) 731.

[5] GASPARD, J. P. and CYROT-LACKMANN, F., J . Phys. C (1972).

[6] CYROT-LACKMANN, F., J. Physique 35 (1974) C4-109.

[7] BREUIL, M., These d'Etat (Paris), 1971.

[8] GASPARD, J. P. and CYROT-LACKMANN, F., J. Phys. C 6 (1973) 3077.

[9] HAYDOCK, R., HEINE, V. and KELLY, M. J., J. Phys. C 5 (1972) 2845.

[lo] ROTH, L. M., Phys. Rev. B 7 (1973) 4321, and J. Physique 35 (1974) 04-317.