MÖSSBAUER EFFECT AND VACANCY DIFFUSION

(1)

HAL Id: jpa-00216602

https://hal.archives-ouvertes.fr/jpa-00216602

Submitted on 1 Jan 1976

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

MÖSSBAUER EFFECT AND VACANCY DIFFUSION

L. Gunther

To cite this version:

(2)

JOURNAL DE Colloque C6, supplkrnent au no 12, Tome 37, Dicembre 1976, page C6-15

L. GUNTHER

Department of Physics, Tufts University, Medford, MA 02155, U. S. A.

R6sum6. - Nous presentons une th6orie dynamique qui a et6 motivk par la necessite d'expli- quer les r6sultats experimentaux recents des spectres Mossbauer de Fe en Cu, Fe en Au, et Fe en Al. La diffusion dans ces systkmes est dominee par le mecanisme de vacances, qui entrafne des correlations entre les sauts successifs. Par consequent, la theorie qui a 6t6 developpee par Singwi et Sjo- lander pour le spectre Mossbauer d'un noyau diffusant n'est pas applicable. En plus, nous intro- duisons l'inverse du spectre Mossbauer normalis6 et Bvalue & une frequence de zero comme moyen utile de comparer les largeurs expbrimentales et thkoriques des spectres.

Abstract. - We present a dynamical theory of vacancy diffusion which was motivated by the need to explain recent experimental results for the Mossbauer spectra of Fe in Cu, Fe in Au, and Fe in Al. Diffusion in these systems is dominated by the vacancy mechanism, which involves strong correlations between successive jumps. The theory developed by Singwi and Sjolander for the Mossbauer spectrum of a diffusing nucleus is therefore not applicable. We, furthermore, introduce the inverse of the normalized Mossbauer spectrum evaluated at zero frequency as a useful means of comparing experimental with theoretical spectral widths.

1. Introduction. - We have developed a dynamical theory of vacancy diffusion in solids [I] which is applicable to the recent controversial experimental results [2] of Knauer and Mullen [3] on Fe in Cu, Sorensen and Trumpy [4] on Fe in Al, and Anand and Mullen [S] on Fe in Au.

Singwi and Sjolander [6] developed a theory (henceforth referred to by SS) of the Mossbauer spectrum of diffusing atoms which is valid when correlations between successive jumps are absent. The spectrum of a single crystal is a Lorentzian with a width which is greater than the natural line width

r

by an amount AT which is proportional to the diffu- sion constant D. Experiments by Lewis and Flinn [7] on Fe in Fe-3

% Si, where the interstitial mechanism

(which is free of correlations) is dominant, are consistent with SS. The vacancy mechanism, which is dominant in the above controversial experiments [3- 51, involves strong correlation effects, so that SS is not applicable. Theoretical approaches [2, 4, 81 towards understanding the results of these experiments rely on modifications of SS which amount to multiplying

A r by a factor f < 1 which reflects the effect of correlations. Such a correlation factor correction has been shown to obtain for the diffusion constant. No such simple correction for correlation effects should be expected to be adequate for the Mossbauer spectrum. In fact, our theory results in a spectrum which is not Lorentzian even for a single crystal 191.

In order to treat correlation effects adequately, it is necessary to study the motion of the diffusing atom for time scales comparable to the mean jump time z,, since these time scales are strongly reflected by the

Mossbauer spectrum. The diffusion constant reflects the behavior only for time scales much larger than 7,.

It is for this reason that the detailed Mossbauer spectrum can potentially provide much more information than the diffusion constant about diffusion processes. In a complete theory, the state of the system is determined by specifying the positions throughout the lattice of the vacancies, the impurities (which are assumed to be Mossbauer atoms) and the host atoms [lo]. Alternatively, we can view each lattice site as existing in one of three states : being occupied by a vacancy, by an impurity, or by a host atom [ll]. Unfortunately, such a complete treatment of correlations leads to mathematical complexities. Instead, we have developed an approach due to Lidiard [12], whose ultimate concern was the diffusion constant. Three assumptions are involved in the approach :

a) the impurity concentration is so low that we

can neglect those states for which two or more impurities are so close as to interact ;

b) the vacancy concentration is so low that we can neglect states for which an impurity has more than one vacancy as a nearest neighbor ;

c) those vacancies which are not a nearest neighbor of an impurity are not correlated with the impurity and are assumed to be randomly distributed throughout a reservoir of host atoms and vacancies. It is assumption c) which fails to fully take into account correlations. Nevertheless, Lidiard obtains a correlation factor in the case of self-diffusion of 0.82, which compares quite favorably into the exact result of 0.80.

(3)

C6-16 L. GUNTHER

According to the above three assumptions, an impurity, located at a particular lattice position n can find itself in one of (z

+

1)-states : Either it has no nearest neighbor vacancies, or it has a single vacancy in one of the z-nearest neighbor lattice positions. The state of the system in this model is specified by the function

which is the probability that an impurity is located at the site n in the state j at the time t . We obtain a set of linear coupled differential equations for Nnj(t), whose eigensolutions are diffusive modes which are specified by a wave vector k and mode index s = 0, 1,

...,

z. The branch s = 0 is the only one for which the decay rate a,@) approaches zero as k approaches zero (corres- ponding to infinite wavelength or diffusion over lalge distances). Analogous to the acoustic branch of pho- nons, this branch alone determines the diffusion over long time scales and hence the diffusion constant itself. All branches are relevant for the Mossbauer spectrum, however.

The outline for the remainder of the paper is as follows : In section 2 we present a mathematical outline of our theory, with expressions for the Mossbauer spectrum of a single crystal and of a polycrystal. The spectrum is expressible in each case as a superposition of Lorentzians, each reflecting a particular mode (k, s = 0, 1,

...,

z). An important feature of our theory is that it allows us to take into account the dependence of the Mossbauer fraction on the nearest neighbors of the Mossbauer atom when the y-ray is emitted. Since our theory doesn't result in a Lorentzian Mossbauer spectrum, it is not necessarily worthwhile to fit experimental results to a Lorentzian in order to compare the results with theory. If we are content with comparing spectral widths, it is advan- tageous to use the inverse of the normalized spectrum evaluated at zero frequency. It is shown in section 3 that this measure of the width is relatively easy to obtain theoretically.

Finally, in section 4 we present a qualitative comparison of our results with experiment and suggestions for future experimental and theoretical work.

2. Theory. - As we showed in the introduction, the state of the system in our model is specified by the function Nmj(t), which is the probability that an impurity is located at the site n in the state j at time t. Changes in this function take place through the inter- change of a vacancy with a host atom or the inter- change of avacancy with an impurity. The four different categories of interchanges which can occur are depicted for a square lattice in figure 1, with the distinct jump rates v,, v,, k,, and k, indicated.

Let us number the nearest neighbor sites from one through four in a clockwise (or counter-clockwise) manner. Further, let c, be the impurity concentration and a j the basis vector connecting the vacancy to the

FIG. 1.

-

Pictorial representation of the four possible interchanges in the case of the square lattice which lead to a change in the probability function Nnj(t). 0 host atom ; @ impurity ;

vacancy.

jth nearest neighbor site. Then the equations governing the time dependence of Nnj(t) are :

(analogous equations for Nn2, Nn3, and Nn4).

In the case of the FCC lattice of the systems of references 13-51, there are twelve nearest neighbors and we obtain thirteen coupled equations of motion for the Nnj(t).

The general solution to these coupled equations can be expressed as

where v is the volume of the unit cell of the lattice, ej(k, s) is the eigenvector of the mode (k, s), and

g(k, s) is arbitrary. The integral is over the entire Brillouin zone.

The Mossbauer spectrum of a single crystal is given by

(4)

The function &(q) is the Mossbauer fraction of an impurity in the state j and q is the wave-vector of the y-ray. For the FCC lattice, the fj,,(q) have the follow- ing form :

1

fj(q) = exp

(

-

I

p2[< u:

>

sin"

+

where

<

u:

>

and

<

ul

>

are the mean square of the component of the displacement of the Mossbauer impurity in the direction of the vacancy and in the direction perpendicular to the plane of the impurity and vacancy (see fig. I), respectively. The angle 9 is the angle between q and the direction perpendicular to the above plane. NnjSj,(t) is the probability function Nnj(t) given the condition that Nnj(t = 0) = 6,06jjr. It is obtained by setting g(ks) equal to <,(ks), the dual [13] of the eigenvector ej.(ks).

For the single crystal we obtain the spectrum

1

r

- 1

I.,(o) =

;

Re

[-

+

as(q)

+

io] x

s 2

which case this average is approximately equal to an average over the Brillouin zone :

where fj(q) is the average offj(q) over the directions of q.

3. An alternative measure of the spectral width.

-

If the experimental data is sufficiently precise and corrections can be made for finite thickness, one might be prepared to fit the entire experimentally obtained spectrum to theory. Still it is easy to see that a log

(An

vs. T-' plot will lie below the SS line,' as was observed in ref. 13-51. In addition, we expect that the plot will not be a straight line but, rather, should have a concave upward curvature. A less ambitious approach is to compare the experimental and theoretical spectral widths, as is usually done. Since the Mossbauer spectrum is not a Lorentzian, there is no reason to fit the spectrum to a Lorentzian. We would like to suggest the use of another measure of the spectral width.

The expression for the Mossbauer spectrum given by eq. (3) is normalized; that is the area under the spectrum is unity :

wave vector - k

FIG. 2. -Schematic drawing of the mode decay rate as&) for the various branches s = 0, 1,

...,

z.

For the polycrystal, we must average over the crystallographic orientations. This is equivalent to averaging over the directions of the wave vector q. Usually q-' is much larger than the lattice spacing, in

In the case of a Lorentzian of width T , the quantity 1(0)

is equal to 2/nr. Furthermore, the quantity I(0) is

much easier to calculate from theory than the half- width. It is therefore quite convenient to measure the width by the quantity

2

y = -

nI(0) (8)

which equals the width at half-maximum of a Lorent- zian. If the experimentally obtained spectrum I,,,(o) is not yet normalized, this width is given by

2(area under I,,,(o)) y =

nIexp(0)

4. Discussion.

-

We have not yet developed a reliable simplified form for the complicated expressions for the Mossbauer spectrum resulting from our theory. A complete numerical calculation will require the jump rates v,, v,, k,, and

k,,

the Mossbauer fractions fj(q), the eigenvectors es(k), and the mode decay rates a,(k). Some of these quantities might be fitted to experimentally obtained Mossbauer spectra. (To fit all of them would be a horrendous job.) The jump rates

(5)

C6-18 L. GUNTHER

could be obtained from diffusion experiments [14]. The Mossbauer fractions &(q) could be obtained from theoretical and/or experimental studies of lattice vibrations in the presence of impurities and vacancies [15]. Finally, when account is taken of the spatial correlations between a vacancy and an impurity due to their interaction, as can be done in the S = 1 model [ll], c, will have to be replaced by the temperature dependent probability for the nearest neighbor of an impurity to be a vacancy.

To date, Mossbauer experiments have been carried out over such small ranges of temperature and with such low precision as to make a detailed comprehen-

sive comparison between experiment and theory quite difficult. The experiments of refs. [3-51 were carried out over such a narrow range of temperature Tthat the fit of log

(Ar)

vs.

T-'

to a straight line would seem fortui- tous. It is hoped that experimentalists will improve upon techniques in both these areas so that a satisfac- tory comparison can be made. This would result in greatly improving our ability to determine the dominant mechanisms responsible for diffusion in specific systems and to better understand the role that vacancy diffusion plays in other types of behavior in solids which are affected by vacancy diffusion over time scales on the order of the mean jump time.

References

[I] GUNTHER, L., Bull. Am. Phys. Soc. 21 (1976) 809. 121 See critique by JANOT, C., J. Physique 37 (1976) 18.

[3] KNAUER, R. C. and MULLEN, J. G., Phys. Rev. 174 (1968) 711.

[4] SBRENSEN, K. and TRUMPY, G., Phys. Rev. B 7 (1973) 1791.

[S] ANAND, H. R. and MULLEN, J. G., Phys. Rev. B 8 (1973) 3112.

[6] SINGWI, K. S. and SJOLANDER, A., Phys. Rev. 120 (1960) 1093.

[7] LEWIS, S. J. and FLINN, P. A., Appl. Phys. Lett. 13 (1968) 150. [8] KNAUER, R. C., Phys. Rev. B 3 (1971) 567.

[9] For a poly&ystal SS leads to an average of Lorentzians over crystaflographic orientations. This average is not a Lorentzian. It is incorrect to average the SS width and assume that the spectrum is a Lorentzian.

[lo] The details of the vibrational motion about the lattice sites are reflected in the background of the Mossbauer spectrum. This background has a width on the order of the Debye frequency and so may be neglected.

[ l l ] We are presently developing a theory which treats the system as a lattice of interacting S = 1 spins.

[12] LIDIARD, A., Phil. Mag. 46 (1955) 1218.

[13] The solution of eqs. (1) leads to a determinant which is not

symmetric.

[14] MULLEN JAMES G., Phys. Rev. 121 (1960) 1649.

[15] See the review of this subject in MARADADUDIN, A. A., MONTROLL, E. W., WEISS, G. H. and IPATOVA, I. P.,