DIVERSCOMPUTER EXPERIMENT ON THE EFFECT OF POINT DEFECTS IN LEAD AT 300 °K

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DIVERSCOMPUTER EXPERIMENT ON THE EFFECT OF POINT DEFECTS IN LEAD AT 300 °K

E. Stoll

To cite this version:

E. Stoll. DIVERSCOMPUTER EXPERIMENT ON THE EFFECT OF POINT DEFECTS IN LEAD AT 300 °K. Journal de Physique Colloques, 1971, 32 (C2), pp.C2-253-C2-256.

�10.1051/jphyscol:1971252�. �jpa-00214578�

DIVERS

COMPUTER EXPERIMENT ON THE EFFECT OF POINT DEFECTS IN LEAD AT 300 OK

E.

STOLL

IBM, Zurich, Research Laboratory, Switzerland

*,

8803 Riischlikon

Rksumk. - Nous avons utilise le modble du potentiel developpe dans une publication precedente pour calculer le potentiel effectif interionique du plomb. Nous avons ajuste les branches de dispersion des phonons par des experiences de diffusion non klastique des neutrons.

A l'aide du procede de Rahman et de notre modble nous avons resolu les equations du mouve- ment de Newton dans le cas de 864 ions de plomb places dans une enceinte presentant des condi- tions aux limites periodiques. En considerant la trajectoire de chaque ion nous avons calculB l'autocorr~lation de la vitesse

<

v(t)v(O)

>

et sa representation de Fourier caractMsCe par la fonction de distribution de frkquences g(w). Nous avons employe le procede de Rahman dans deux cas ; dans le premier nous avons considere un reseau parfait, dans le second un ion du reseau a kt6 enleve.

Les resultats montrent que :

1. Prbs de la lacune les ions se deplacent autour de leur ancienne position d'equilibre ou tombent dans la lacune. C'est le point de depart de la diffusion des trous. Le coefficient de diffusion est alors plus grand que le coefficient de diffusion extrapole du plomb liquide. Ainsi la densite maximale des lacunes dans le plomb est une fraction de l'ordre de grandeur d'une lacune pour 1000 ions.

2. Une analyse detaillee de la fonction de distribution des frkquences Ag(co) de la difference entre le systeme affecte d'un defaut ponctuel et celui se trouvant en Ctat parfait montre que les pulsations radiales apparaissent dans les couches formees par les ions places autour de la lacune et d'autre part que les couches voisines se deplacent en direction opposee. Cette longueur tres courte d'onde provient de la breve periode de temps pendant laquelle la lacune reste en un lieu quelconque.

I1 en resulte que les ions se deplaqant et mis en mouvement par la diffusion des trous se compor- tent exactement comme des particules libres dans le champ produit par les autres ions.

Abstract. - Using the model potential method developed in an earlier paper we have calculated the effective ion-ion potential in lead by fitting the phonon dispersion branches of inelastic neu- tron diffraction experiments. With the Rahman process and the above-mentioned ion-ion potential we have solved Newton's equation of motion for 864 lead ions in a box with periodic boundary conditions. Following the path of each Pb-ion we calculated the velocity-auto-correlation

<

v(t) v(0)

>

and its Fourier representation characterized by the frequency distribution function g(w). The Rahman process was calculated twice, once for a perfect lattice and a second time for a lattice with a single ion removed. Results show that :

1. Near the vacancy, ions move around the old equilibrium position or fall into the empty loca- tion. T h ~ s is the starting point of a hole diffusion. The calculated diffusion coefficient is an order of magnitude larger than the extrapolated diffusion coefficient of liquid lead. Thus the maximum density of vacancies in lead is at least one order of magnitude less than one vacancy per 1 000 ions.

2. Detailed analysis of the frequency distribution function Ag(w) representing the difference between perturbed and unperturbed system, shows that the radial pulsations are set up in the shells formed by the ions surrounding the vacancy and that neighboring cells move in opposing directions. This short-wavelength motion is caused by the brief period of time that the vacancy spends at any one location. Ions travelling due to hole diffusion move as free particles in the field of the other ions.

I. Introduction. - The progress made inund erstand- ing the nature of the effective interatomic interaction and the development of fast and large computers ena- bles us t o solve Newton's equation of motion in clas- sical systems of condensed matter. Rahman

[I,

21 and Verlet [3] developed recursive methods for sol-

*

Part of the work was performed at the Swiss Federal Institute for Research, Wiirenhingen, Switzerland, prior to the author joining IBM.

ing the Newtonian equations in a box containing

a

large number of classical particles. In their well- known papers they succeeded in explaining a numbre of dynamical and static effects of liquid argon. On the other hand, Dickey and Paskin

[4]

as well as the author [5] have used the Rahman process to obtain dynamical and related properties of solid argon and lead. Employing this powerful method for solid lead at room temperature with a point defect produced by a missing Pb-atom in the box mentioned above we

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971252

C2-254 E. STOLL

expect to gain a better understanding of the dynami- cal properties due to this defect.

The next section contains a brief description of the method. The following two deal with the nature of the motion of a single atom around the vacancy and with hole diffusion. These sections also contain an estimate of the point defect density a t room temperature and describe the nature of the collective motion of atoms due to the defect. Section V contains concluding remarks.

11. Brief description of the method. -

Using the model-potential-method [6] we have developed an electron-ion model-potential in lead by fitting the phonon dispersion branches to the inelastic neutron diffraction measurement of Brockhouse et al.

[7].

With this model we have calculated the effective ion- ion potential @(R) between two Pb-ions in the solid lattice. This potential was then introduced into New- ton's equations of motion (1)

:

Here M i stands for the mass and Xi for the vector coordinate of the i-th ion, t denotes time, X i j is the difference vector coordinate, R i j is the distance between the i-th and the j-th ion, and R is the distance variable.

The ions are enclosed in a box satisfying periodic boundary conditions. Hence its form is limited by the requirement of translational invariance. The dimension of each basis vector of the box is given by a triplet of integers related to the basis vectors of the unit cell in the solid. This limitation severely restricts the number of possible excitations as well as the accuracy of the method. On the other hand, the condition renders each

ion equivalent to any other. Thus we are completely free to choose the ion we want to remove in order to form a point defect. For the shape of the box we choose a cube with side length equal to six unit cells of the fcc Pb lattice. The box contains 864 ions in the unper- turbed system and 863 ions in the perturbed one.

Rahman 123 developed the vector coordinates Xi(t) of each ion by means of their first 6 derivatives. Using these in a Taylor series (eq. (2)) he obtains estimates for the vector coordinates and their 6 derivatives at t + ^At,

In a next step he compares the second derivatives obtained from Newton's equation ( 1 ) with those in the estimate (2). The difference vector Ai for each atom, eq. (3), is inserted into a new equation, eq. (4), in order to correct the estimates.

A. =

a2xi(t + ^{a t )} IN ^{- a2xi(t + a t )} / ^(,)

at2 at2

e

ajxi(t + ^{a t )}

^- -

^ajxi(t + ^{a t )} I + f j Ai .

atj at'

e

This transforms the problem of solving Newton's equations into a feedback process, where At is the feedback parameter. Taking At small enough the system is stable at all times.

111. Motion picture and hole diffusion. -

The motion picture of 4 Pb ions is shown in figure 1 by projecting their paths onto the (001) plane. Circles denote mean amplitude of motion for each ion. The

x

t, = 1.25 t, = 2.50 psec

o t,

= 3.75

t5 5.00

FIG. 1. - Projection onto the (001)-plane of the paths of motion around a vacancy of 4 Pb nuclei. Circles denote mean ion

amplitudes of motion.

COMPUTER EXPERIMENT ON THE EFFECT OF 'POINT DEFECTS IN LEAD AT 300°K C2-255

ions on the left and in the lower corner on the right

move around their equilibrium positions. Their adherence to this position is rather unexpected. In contrast, the ion in the upper right-hand corner has left its equilibrium position and fallen into the empty hole in the middle of the picture. In the extremely short time of only 4 ps the hole has diffused between two equilibrium positions, and we therefore expect a large diffusion coefficient. Calculating the position auto-correlation we find a diffusion coefficient of about 7.2 x cm2 s-'. This value is extremely large compared with the value of 0.64 x loT7 cm2 s-' extrapolated from liquid lead. Hence we can estimate that the density of point defects is at least one order of magnitude smaller than one defect in 1 000 ions at room temperature.

IV. Collective excitations.

-

To answer the ques- tion of how a point defect influences the collective excitation spectrum we analyze the velocity distribu- tion. The mean square radial velocity component in a system containing a vacancy is about 1.4 % larger than in the unperturbed lattice. We have to expect a pulsating motion of the ion shells surrounding the hole. It should be noted that this motion is superposed on the normal modes of the unperturbed part of the system. Hence we expect it to have a small effect on the frequency distribution function g(o) in figure 2.

FIG. 2. - Frequency distribution function g(w) of solid lead at 300 O K with a point defect.

We obtain this function by Fourier-transforming the velocity-auto-correlation of all Pb-ions in the box.

To avoid break-off effects the distribution function is folded with a Gaussian possessing a half-width of 0.6 THz. On the left at 5 THz there appear the contri- butions of transverse phonons and on the right at 13 THz those of longitudinal ones. As we expected, the curve exhibits few details. In contrast to this, the frequency distribution function Ag(w) of figure 3, characterizing the frequency difference between per- fect and perturbed systems, contains a great deal of information. On the right are two separated lines, representing longitudinal phonon contributions. Com- pared with the phoilon dispersion curves calculated with the same model-potential these lines correspond

to the contribution of phonons originating at the boundaries of the Brillouin zone. Hence the physical interpretation we can give to these lines is that pulsat- ing shells tend to move essentially in opposing direc- tions. In the central portion of the curve we find a further line. Remembering the definition of the Ein- stein frequency O, we find that an ion diffusing from one equilibrium position to another oscillates like a free particle in the field of the other ions. The

<<

nega- tive

⁾⁾

lines at the left occur because transverse oscilla- tions are relatively weak compared to longitudinal ones.

FIG. 3. - Frequency distribution function Ag(w) characterizing the difference between perturbed and unperturbed solid lead at 300 OK.Qo denotes the Einstein frequency and the two short

bars indicate its uncertainty.

Calculations were not extended beyond the short time of 5 ps because the diffusion of the hole destroys the accuracy won by extending the length of Fourier- transform integration. The extremely short pulse duration makes it impossible to create long-wave- length phonons. This is the reason why neighboring shells pulsate in opposite directions.

V. Conclusions. -

Computer experiments are a powerful method for explaining the effect of point defects. The hole diffusion is observable and a possible density of defects at room temperature in solid lead can be estimated. The diffusing ions move as free particles in the field of the other ions. The shells around the vacancy pulsate one against each other.

The broad line width of this motion is caused by the hole diffusion and does not allow shell motions other than the one mentioned above.

Acknowledgements. ^-

I am grateful to Prof.

A. Rahman for sending us his Fortran-program, and

I should like to thank him and Dr. T. Auerbach for

stimulating and helpful discussions. I acknowledge

with pleasure the cooperation and assistance of the

computation centers of the Swiss Federal Jnstitute

for Reactor Research and of the Swiss Federal Insti-

tute of Technology, where calculations were carried

out.

References

[I] RAHMAN (A.), Phys. Rev.. Letters, 1967, 19, 420. [5] STOLL (E.), Helv. Phys. Acta, 1970, 43, 456.

[2] RAHMAN (A.), Phys. Rev., 1964, 136, A 405. [6] SCHNEIDER (T.) and STOLL (E.), Phys. kondens. Materie, 1966, 5, 364.

[3] VERLET (L.), Phys. Rev., 1967, 159, 98. [7] BROCKHOUSE (B. N.), ARASE (T.), CAGLIOTI (G.), [4] DICKEY (J. M.) and PASKIN (A.), Phys. Rev., 1969, RAO (K. R.) and WOODS (A. D. B.), Phys. Rev,

188, 1407. 1962, 128, 1099.