DETERMINATION OF LOCAL ATOMIC ORDER USING THE MÖSSBAUER EFFECT

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DETERMINATION OF LOCAL ATOMIC ORDER USING THE MÖSSBAUER EFFECT

L. Schwartz, A. Asano

To cite this version:

L. Schwartz, A. Asano. DETERMINATION OF LOCAL ATOMIC ORDER USING THE MÖSSBAUER EFFECT. Journal de Physique Colloques, 1974, 35 (C6), pp.C6-453-C6-457.

�10.1051/jphyscol:1974694�. �jpa-00215851�

LOCAL ATOMIC ORDER I N ALLOY SYSTEMS.

DETERMINATION OF LOCAL ATOMIC ORDER USING THE MOSSBAUER EFFECT (*)

L. H. SCHWARTZ and A. ASANO (**) Materials Science Department

Northwestern University, Evanston, Illinois 60201, U. S. A.

Rksum6.

Dans cet article les auteurs font une analyse critique des articles recents relatant la mesure des paramktres d'ordre

courte distance

de Warren, a partir du spectre Mossbauer d'alliages ferreux ferromagnetiques. 11s montrent que la fraction w(i, j) d'atomes de fer avec des atomes de solute i- en premiers voisins et j- en seconds voisins peut Btre mesurke avec une precision suffisante, mais seulement pour un nombre limit6 de valeurs w(i, j). La determination precise des w(i, j) nkcessite la correction due a I'effet d'une epaisseur finie et l'identification du champ hyperfin caracteristique de la configuration (i, j) particulikre. Dans certains cas cette derniere obligation ntcessite l'utilisation d'bchantillons monocristallins pour I'identification des interactions aniso- tropiques hyperfines. De plus, ils montrent que le nombre limit6 de w(i, j) qui peuvent 6tre mesures, n'est pas suffisant pour la determination des ai. Cependant lorsque des mesures des

sont obtenues par diffraction, une technique numkrique peut 6tre employ& pour obtenir les w(i, j). 11s presentent des exemples pour des alliages dilues de Mo dans le fer pour lesquels les rbultats d'effet Mossbauer et de diffraction des rayons X ont kt6 obtenus sur des monocristaux.

Abstract.

In this study a critical analysis is made of recent papers purporting to have measured the Warren short-range order parameters, ai, from Mossbauer spectra of ferromagnetic iron alloys.

It is shown that the fraction of iron atoms with i- near neighbor and j- next-near neighbor solute atoms, w(i, j), may be measured with considerable accuracy, but usually only for a limited number of the (i, j) values. The accurate determination of the w(i, j) requires correction for the effects of finite sample thickness and the identification of the hyperfme fields characteristic of the particular (i, j) configuration. In some cases this latter requirement necessitates the use of single crystal samples to identify anisotropic hyperfine interactions. It is further shown that the limited set of w(i, j) which can be measured are not sufficient to determine the

However, when the

have been measured by diffraction, a numerical technique can be used to obtain the w(i, j). Examples are presented for dilute alloys of Mo in iron for which both Mossbauer and X-ray data have been obtained from studies of single crystals.

Introduction.

From the earliest days of their awareness of the Mossbauer effect, metallurgists a n d solid state physicists have exploited the sensitivity of the hyperfine interactions t o local environment in studies of atomic arrangements in solids. Notable examples of early quantitative studies of long-range atomic order are the estimates of the degree of order in Fe-Rh alloys by Shirane et al. [I] a n d the prediction of CsCl type ordering in Fe-V alloys by Preston et al. [2]. More recent attempts t o extract the Warren short-range order parameters, ai, from the Mossbauer spectra of ferromagnetic iron alloys, can be traced t o the paper of Heilmann and Zinn [3]. While approxima-

tely valid in the case of nearly complete order, the analytical approach of Heilmann and Zinn contains serious errors which preclude its application t o alloys with partial long-range o r short-range order a s was done in the case of Fe-12.3 at. pct. A1 considered by Briimmer et al. [4]. This .paper examines the experi- mental a n d analytical problems associated with extract- ing information about local atomic arrangements from the Mossbauer spectrum. As an example of what can be expected from this technique, a comparison is made of recent measurements of the local atomic arrangement in Fe-3.2 at. pct. M o using both X-ray diffuse scattering and Mossbauer spectroscopy 151.

1. Experimental. - 1 .1 HYPERFINE

(*)

This research represents a portion of the thesis submitted - ~h~ basis for most of the analyses of ~ o ~ ~ b ~ ~ ~ ~ by A. Asano for the Ph. D. degree in Materials Science, and was

sponsored by ARPA and NSF through the Materials Research 'pectra obtain hyperfine in iron Center at Northwestern University. is the work of Wertheim et al. [6]. The Mossbauer

(**)

Present address

1-10-9 Matsugaoka, Kugenuma, Spectrum for dilute iron alloys was shown by these Fujisawa City, Kanagawa Pref., Japan. workers t o be due t o the superposition of many six line

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974694

C6-454 L. H. SCHWARTZ AND A ASANO

sub-spectra, each with hyperfine magnetic fields and isomer shift determined by the near neighbor (n. n.) and next-near neighbor (n. n. n.) solute concentration.

The intensities of each of these sub-spectra was taken proportional to the fraction of iron atoms with i - n. n. and j - n. n. n. solute atoms, w(i, j). Wer- theim et al. concluded that the hyperfine magnetic field H(i, j) at a given iron nucleus was changed from that for FeS7 in iron metal, H,, by an amount

H(i, j)

Ho(l + ^kc)

^H,(ia + jb) (1 + kc), (1) where a, b, and k are empirically determined for a particular alloy and c is the atomic concentration of the solute. The essential sufficiency of eq. (1) in describing various iron alloys has been verified by many workers (see, e. g. Vincze and Campbell [7]) ; however, Stearns [8] has shown the importance of considering the weak effects of solute atoms more distant than the second neighbor shell in determining the origin of the hyperfine fields.

The most serious shortcoming of eq. (1) in the present context is the assumption that the magnetic hyperfine field is isotropic, i. e. that H(i, j) is determined only by the distance of the solute atom from the iron atom in question. Polycrystalline samples are not useful for delineating anisotropic effects as the sample averages over all crystallographic orientations. Cranshaw [9]

using single crystal samples has shown that strong anisotropic perturbations to the magnetic hyperfine field exist in many alloys of iron with transition and nontransition elements, and in particular for the Fe-3.2 at. pct. Mo alloy discussed below, Asano and Schwartz [lo] have quantitatively determined this effect using single crystal samples. This anisotropic interaction partially eliminates the degeneracy for iron atoms in the same neighbor shell, splitting the reso- nance line associated with these iron atoms. The amount of the splitting depends on the angle between the magnetic spin polarization and the vector from the iron to solute atom, and differs in magnitude for diffe- rent solutes. In the case of iron rich Fe-Mo alloys, ahen the spins are aligned along the easy axis [I001 (as in all experiments on samples in the absence of magnetic fields), the anisotropic effect splits the resonance line associated with iron atoms with one Mo in the n. n. n.

shell by an amount of the order of AH(0,1)/2 as shown in figure 1. The 0's indicate the data taken in trans- mission through a 1 mil thick single crystal absorber of Fe-3.2 at. pct. Mo. Only the lowest energy resonance envelope is shown. In figure l a the data have been fit as is most commonly done, neglecting the anisotropic interaction. In figure lb the anisotropic splitting has been accounted for (for details of the fitting pro- cedure, see ref. [5] and [lo]). Neglect of this splitting in the analysis can lead to an erroneous determination of all the w(i, j) and hence to an error in the estimate of the local atomic order in the sample. The magnitude of the error in this example will be discussed below in reference to table I. While the magnitude of this aniso-

I

-6.00 - 5.00 -4.00

VELOCITY

( m m / s )

FIG. 1. - The lowest energy envelope of the Mossbauer spec- trum of single crystal Fe-3.2 at. pct. Mo taken in transmission with an external magnetic field of 2.7 kOe along the [I001 direction. 0-0 : experimental data

: (0,O) peak

.

(1,O) peak

(0,l) peak

...

superposition of the (1,1), (2,0), (0,2) peaks

a) Analysis neglecting anisotropic hyperfine interaction. b) Analysis including anisotropic hyper- fine interaction. No correction for finite sample thickness

has been made.

tropic effect is probably negligibly small for dia- magnetic solutes [8] and for transition metal solutes near iron in the periodic table, it is clearly not negligible for transition metal solutes far from iron [9].

1 . 2 INTENSITY

The common procedure for fitting complex Mossbauer spectra is to determine the best fit to the data of a linear superposition of Lorentzian absorption lines. This approach neglects the effects of thickness distortion due to finite sample thickness, an effect described many years ago by Mar- gulies and Ehrman [l l]. Many techniques for correct- ing the data for this effect have been proposed, and several of these were reviewed recently by Lin and Preston [12]. For thin absorbers, this correction may be small, but it can never be completely neglected in the determination of the w(i, j).

A convincing demonstration of the importance of

making a thickness correction is illustrated using the

data taken by Asano [5] for Fe 3.2 at. pct. Mo. The

sample was 1 mil thick single crystal examined in trans-

mission. The thickness correction was made by decon-

voluting the observed spectrum (using a technique

developed by Asano [5, 101) to obtain the true absorp-

tion cross-section. In the hyperfine field analysis,

DETERMINATION OF LOCAL ATOMIC ORDER USING THE MOSSBAUER EFFECT C6-455

Measured and Calculated Values of w(i, j) for Fe-3.2 at. pct. Mo

Mossbauer

no corrections 0.61 0.20 0.15 0.003 0.001

Mossbauer

without thickness

correction 0.55 0.26 0.12 0.06 0.001

Mossbauer

all corrections 0.63 0.185 0.115 0.06 0.001

Random, calculated from eq. (3)

with P I = P ,

c 0.64 0.165 0.125 0.06 0.001

X-Ray, calculated from Gehlen-

Cohen-Gragg Program [I 5, 161 0.67 0.12 0.09 0.11 0.001

account was taken of the large anisotropic hyperfine magnetic field perturbation. The results for w(i, j) are presented in table I. The values given on the first line represent the best fit when no corrections have been made, corresponding to the fit shown in figure la. The second line shows the effect of including the anisotropic hyperfine field interaction as shown in figure lb, but neglecting the thickness correction. The third line represents the best analysis of the data, including cor- rections for anisotropic fields and sample thickness. A comparison of these numbers to those shown in the fourth line (calculated for a random distribution of the solute atoms as described below) is quite revealing.

When properly treated, the data agree within experi- mental error with a random solute model. Failure to correct for the thickness distortion leads t o an erro- neously low w(0, 0) and high w(1, 0), a result which would be interpreted as a preference for unlike n. n.

pairs, or short-range order.

By chance in this particular alloy, failure to make any corrections at all would lead to values of w(i, j) rather close to those obtained from the corrected data, but one should not expect this to be generally true. The last line in table I was obtained from the results of an X-ray diffuse scattering study of this same alloy, and will be discussed below.

2. Analytical. -The study of atomic order in a binary alloy by diffraction leads to the determination of the conditional probabilities pAiB(ri), the probabili- ties of finding a B atom at the end of vectors ri when an A atom is at the origin. These quantities are often abbreviated pfIB, where the subscript i refers to vector ri, and interpreted as the probabilities of finding a B atom at any one site in the ith neighbor shells of an A atom. The diffraction results are usually expressed as a set of Warren short-range order parameters [13], a,, defined by the relation

It is important to emphasize that pAIB(ri) is a condi- tional or pair probability (the discrete analogue of the

pair density function in liquids), and desc~ibes only the fraction of pairs of atoms separated by the vector ri sta~ting on an A and ending on a B. Thus p?lMo gives the probability of finding a Mo atom at a particular site in the n. n. shell of an Fe atom, but it tells us nothing about the simultaneous occupation of the remaining seven sites in the n. n. shell. It is not correct to assign the probability

to each of these sites simultaneously as suggested by Heilmann and Zinn [3]

and by Briimmer et al. [4], as this is equivalent t o assuming that the probability of finding an Mo at each of the sites in the n. n. shell is independent of the occupation of all other sites in the shell.

The analysis of Heilmann and Zinn continues with the use of a binomial probability function to calculate w(i, j), i. e.

where z , and z, are the numbers of n. n. and n. n. n.

sites, respectively. In the derivation of eq. (3), it is assumed that the quantity P I (or P,) is the probability of finding a solute atom at a specific site in the n. n.

(or n. n. n.) shell and that this probability is the same for each n. n. (or n. n. n.) site. Thus, what is required is a site probability, not the conditional probability p?IB (or &IB). In fact, for an alloy with no long-range order the site probability for every site is the atomic fraction c regardless of the degree of local order. Clearly with PI

P, = c, eq. (3) does give a correct evaluation of w(i, j) for a random atomic arrangement.

When the alloy has extensive long-range order the

different neighbor shells may be classified as belonging

to different sublattices of the ordered superlattice. In

this case, the site probability may be written for each

sublattice in terms of the long-range order parameter S

for sites far away from the origin atom. However, as

shown for example by Schwartz and Cohen [14] in an

X-ray study of partially ordered Cu,Au, the short-

range order parameters for n. n. and n. n. n. pairs may

differ considerably from the values derived from the

C6-456 L. H. SCHWARTZ AND A. ASANO

long-range order parameter

i. e. the disorder is not random. Thus even for partial long-range order it is not correct to set Pi

for n. n. and n. n. n. shells.

3. Conclusions. - In summary, the analysis of the Mossbauer spectra of ferromagnetic iron alloys can yield a limited number of the set of values w(i, j) ; however, there is no known analytical way to derive the w(i, j) from the Warren short-range order parameters a,. h he reverse process would be possible if the entire set of w(i, j) could be determined, since, for example for then. n. shell, p:lB

ii,/z,, where Ti,

iw(i, j).

i . i

However, the large shifts in hyperfine fields for i + j 2 3 preclude the separation of these subspectra from other lines in the pattern, and even though those values of w(i, j) may be small, they are strongly weighted in the calculation of 5.)

Fortunately, there are numerical techniques for obtaining the w(i, j) from a known set of a,. Gehlen and Cohen [15] have developed a technique for the computer simulation of an atomic arrangement con- sistent with the measured concentration

long- range order parameter S, and short-range order parameters, ai. This program has been extended and discussed extensively by Gragg [16]. No energetic model is assumed. The atoms are merely rearranged in the computer to satisfy the geometrical requirements of the measured order parameters. From such a simu- lation, one may obtain pictures of the atomic arrange- ments, or for the present problem, obtain the desired w(i, j) by a simple counting procedure. An alternative procedure for obtaining this information is the statis- tical mechanical approach of Clapp [17]. In this approach a pairwise interaction relationship between the short-range order parameters and the configura- tional energy is assumed, and the most probable distri- bution of multisite configurations is numerically cal- culated. In this analysis computational difficulties have limited the available information t o w(i)

w(i, j),

J

but for systems in which the hyperfine fields ale pri- marily influenced by n. n. solute atoms, this should suffice.

The Gehlen-Cohen-Gragg program allows one to see the magnitude of the error associated with calculating the w(i, j) from eq. (3) with Pi = p f l B as suggested by Heilmann and Zinn. The case of Fe-6.1 at. pct. Mo will be considered, using the measured short-range order parameters [18] a ,

0.069, a,

0.042, a ,

0.024. Using eq. (3) one obtains the results listed

in the first line of table I1 for the fraction of Fe atoms with zero, one, two, and greater than two Mo in the first two neighbor shells. By contrast, when the Gehlen- Cohen-Gragg program is used with these measured ^ai, the results given in the second line of table I1 are obtained. Note, for example, that an error of 44% is made in the ratio w(l)/w(O) by using the incorrect analysis in this alloy which exhibited only relatively weak clustering (preference for Fe-Fe and Mo-Mo pairs).

Calculated Values of w'(1) or Fe-6.1 at. pct. Mo

w'(0) w'(1) w'(2) w' (I

- - -

Calculated from eq. (3) with P I from eq. (2),

a1 =

0.069 0.436 0.373 0.148 0.043 Calculated from Geh-

len-Cohen-Gragg Pro-

gram [l5, 161 0.498 0.296 0.133 0.073 w'(1)

w(i,

fraction of Fe atoms with 1 Mo atoms

i ,j

in then. n. and n. n. n. shells.

i + j = l

Returning to table I, the results of an X-ray diffuse scattering analysis of Fe-3.2 at. pct. Mo [S] giving a,

0.056, a,

0.059, a ,

0.01 3, have been convert- ed to the w(i, j) listed in the last line using the Gehlen- Cohen-Gragg program. Within the experimental errors of the two techniques, the X-ray and Mossbauerresults are equivalent to the assumption of random solute distribution as seen by comparing the figures in the last three lines of table I. It is worth noting that for this particular composition, a change in a , of 20 pct.

results in a variation in w(1,O) of only 10 pct.

In conclusion, it may be said that while the Moss- bauer effect cannot be used to directly obtain the Warren short-range order parameters a,, it can be used to obtain an interesting and unique measure of the local order, the set of numbers w(i, j). When combined with the Gehlen-Cohen-Gragg technique or that of Clapp, one may use the measured w(i, j) as a unique check of the short-range order measured by diffraction techniques.

Acknowledgments.

The authors wish to acknow- ledge the valuable assistance of Professor J. B. Cohen, in both the measurement and the analysis of the X-ray diffuse scattering.

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