Sites are Separable in Garnets with ALCHEMI

JaÂnos L. LaÂbaÂr1;�, Jerzy Morgiel2, Lajos ToÂth1, and IstvaÂn DoÂdony3

1 Research Institute for Technical Physics and Materials Science, Budapest, Hungary2 Institute of Metallurgy and Materials Science, Krakow, Poland3 EoÈtvoÈs LoÂraÂnd University, Budapest, Hungary

Abstract. It is shown in this paper that ± in contrast to

the accepted belief in the literature ± it is possible to

determine if a minority component is located on the

dodecahedral, octahedral or tetrahedral sites in a garnet

single crystal. Previous literature regarded the dode-

cahedral sites indistinguishable form the tetrahedral

sites. Our prediction about the separability is based

upon dynamical Bloch-wave calculations and proved

experimentally in a transmission electron microscope

(TEM) on two different natural garnets (almandine and

grossular). The crystallographic positions of the min-

ority components are determined in an ALCHEMI

experiment that makes use of the channeling of elec-

trons in special directions of a single-crystal sample.

Key words: Crystallographic site separation; TEM; ALCHEMI;channeling; garnet.

The garnet structure is that of an oxide with space

group Ia3d with the general formula X3Y2Z3O12,

where X, Y and Z denote the dodecahedral, octahedral

and tetrahedral sites in the lattice, respectively. These

sites are occupied by divalent, trivalent and tetravalent

elements, respectively. Only the occupations of the

octahedral (Y) sites were measured in the literature

dealing with ALCHEMI of garnets because it seemed

to be a generally accepted assumption that the

dodecahedral (X) and tetrahedral (Z) sites are

indistinguishable from each other.

The method Atom Locations by Channeling

Enhanced Microanalysis [1] (ALCHEMI) deduces

the position of a minority component in the unit cell

from the magnitude of the change in X-ray intensities

in response to the change in the crystal orientation

relative to the electron beam. When a single Bragg-

re¯ection is excited, we speak of planar ALCHEMI

[2] and when the sample is examined at and close to a

low index zone axis, the experiment is called axial

ALCHEMI [3]. The general aim of ALCHEMI is

to determine the so called occupation numbers

( fi, j, i � X, Y, Z) (i.e. the partitioning of a minority

component `j' between the three individual sublattices

`i'). Although it is obvious that for every element, `j'

and sublattice, `i' these numbers must be

0 � fi; j � 1 andX

i�X; Y; Z

fi; j � 1; �1�

unphysical values are also published sometimes [4],

most probably originating from two sources. First,

from the unreliability due to largely scattered

measured data. Second and most importantly, for-

mulae that are only valid for planar ALCHEMI (i.e.

for two independently contributing sites) were used in

an axial channeling situation. The assumption that the

X-sites can not be distinguished from the Z-sites

served as a basis to justify this type of evaluation. This

assumption is valid [5] for planar ALCHEMI but it is

invalid for axial ALCHEMI in the direction of the

h111i zone axis, as we show it in this paper. It is

proved in this paper that the 3 sites can only be

distinguished experimentally, if a formula with 3

independent internal standards is used in the course of

quanti®cation.

Experimental and Calculation

The Bloch states corresponding to the energetic (200 keV)electrons of the beam of the TEM inside a thin single crystal

Mikrochim. Acta 132, 489±492 (2000)

� To whom correspondence should be addressed

garnet (almandine, i.e. (Fe3Al2Si3O12)) are calculated using theEMS program [6] of Stadelmann. The distribution of the integratedelectron intensity is projected onto the unit cell (in a planeperpendicular to the electron beam). Only the asymmetric part ofthe unit cell is used, the rest can be obtained by symmetryoperations. Integrated electron intensity means the intensity of thebombarding electrons integrated through the thickness of thesample. Fig. 1 is an example of the output of such a calculation.The three sites can clearly be distinguished in the [111] projection(Fig. 1). Numerical values of these integrated intensities at theindividual atomic sites were summed up for the equivalent atomicpositions, giving unique integrated values for the X-, Y-, and Z-sites for the given orientation and thickness. The number ofionizations on a given site is proportional to these integratedelectron intensities and the generated X-ray intensities areproportional to the number of ionizations. Such calculations wererepeated for slightly different orientations (by subsequently tiltinginto two directions for a constant sample thickness) and fordifferent sample thicknesses (for the exact h111i orientation) to getthe variation of ionizations with both orientation and thickness(Figs. 2±4 consolidate data originating from these calculations.)Dependencies of the intensities on both thickness and orientationare different for the three sites in almandine, suggesting that theyshould be separable in an ALCHEMI experiment.

Two natural garnets were selected for the experiments thatcontained the same elements (Ca, Fe) but the roles of the sameelements are known to be different in these two samples. Ingrossular [Ca3Al2Si3O12], Ca is a main component, known to sit onthe dodecahedral [X] site, while Fe is a minor component whosecrystallographic position is to be determined. In almandine[ Fe3Al2Si3O12], Fe is a main component, known to sit on thedodecahedral [X] sites and Ca is a minor component whosecrystallographic position is to be determined. Although the role ofthe elements Ca and Fe seems to be completely symmetric in thesetwo experiments, they occupy different crystallographic positionsby substitution in these two samples. Namely, Ca substitutes Fe in

Fig. 1. Projection of integrated electron intensity onto theasymmetric unit of the unit cell of almandine examined from thedirection of the [111] zone axis with 200 kV electrons

Fig. 2. Variation of the number of ionizations at the individualcrystallographic sites of almandine [111] direction as a function ofsample thickness

Fig. 3. Variation of the number of ionizations at the individualcrystallographic sites of almandine close to the [111] direction as afunction of orientation. Tilt is in the direction of the (1ÿ 1 0)reciprocal lattice point

Fig. 4. Variation of the number of ionizations at the individualcrystallographic sites of almandine close to the [111] direction as afunction of orientation. Tilt is in the direction of the (ÿ1ÿ1 2)reciprocal lattice point

490 J. L. LaÂbaÂr et al.

almandine, in accordance with expectations based on the fact thatsolid solution can be formed between almandine and grossular asend-members. So, Fe and Ca both sit on X-site and Fe is divalent inthis series. However, it is not Ca, but Al that is substituted by Fe ingrossular, so Fe occupies Y-site while Ca retains its X-site in thisexperiment. It is in accordance with the expectation that grossularcan also form a solid solution with another garnet, andradit[Ca3Fe2Si3O12]. Fe is trivalent in this series. This knowledge of thesamples also supports that the experimentally determined occu-pancy factors of ours are correct.

The measurements were carried out with a NORAN HPGe (highpurity germanium) detector in a Philips CM-20 TEM operated at200 keV. The strength of the channeling effect was characterizedby the change of the Al/Si ratio from its value measured at arandom orientation.

Results and Discussion

The occupancy factors were determined from simulta-

neous evaluation of 7 spectra collected from the same

sample position with slightly different beam orienta-

tions (i.e. different channeling conditions). By collect-

ing all the spectra from the same part of the sample,

both sample thickness and self-absorption effects were

kept identical during the collection of the entire set of

spectra. Data reduction is based on the assumption that

the emitted X-ray signal of an element is proportional

to the ionizing electron current on the position of the

atom. All three (X, Y and Z) site types are available for

an arbitrary minority component, so three current

values (present at the X, Y and Z sites, respectively)

can contribute to its ionizations. Major components

known to occupy the X, Y and Z-sites respectively

were used as internal standards (i.e. the change in their

X-ray intensities was used to monitor the change in the

currents at the given sites):

Ru; k � fX; u � RX; k � fY; u � RY; k � fZ; u � RZ; k �2�where fi, u (i � X, Y, Z) refers to the fraction of the

given unknown element (identi®ed by the subscript

`u') occupying the X, Y and Z sites, respectively and

Rj; k � Nj; k

Nj; ref

� j � u; X; Y; Z � �3�

represents the enhancement factor, i.e. the ratio of the

intensity of a selected element in the kth spectrum to

the value of the same intensity in the reference

spectrum (collected under non-channeling con-

ditions)�. The four R-values (per spectrum-pair) are

measured from our spectra while the three f-values

(identical for all spectra collected from the same

material) are to be determined from eq. (2). Deviation

of the sum at the right side of eq. (2) from the value at

left side is minimized (simultaneously for the 7

spectra) by varying the assumed values of the three

fi, u. Since precision is limited by the number of counts

in the peaks in the individual spectra, there is no point

in trying to determine the occupancy factors with

better than 1% accuracy. Consequently, a very limited,

®nite number (100 steps for each of the 3 sites � 106)

of values are physically meaningful (see equation (1)).

These physically acceptable values were determined

by minimizing the sum (for the 7 measurements) of

differences between calculated (right side) and

measured (left side) values. Example of data

measured from one of the spectra is presented in

Table 1.

This procedure resulted in { fX, Mn � 0.95,

fY, Mn � 0.05, fZ, Mn � 0.} and { fX, Ca � 0.98,

fY, Ca � 0.02, fZ, Ca � 0.}. Statistical analysis shows

that the precision of the determined occupancy factor

(at the 97% con®dence level) is about 5%. So, we

deduce that these minority elements substitute into the

dodecahedral X-site and the measured small deviation

from it is most probably due to the error in the

measurement. Separation of the X- and Z-sites is

unambiguous [8]. This observation is in accordance

with the anticipated behavior and supports our

prediction.

Similar experimental results (with identical data

reduction) on grossular with Fe content proved that Fe

occupies the Y-sites in this garnet { fX, Fe � 0.,

fY, Fe � 1.0, fZ, Fe � 0.}, in accordance with the

literature. Precision was estimated to be 3% in these

latter experiments.

� A similar formula was used for empirical, analytical ®tting forother structures with the addition of an empirical (physicallyunsupported) term to the right side of equation (2) in [7]. We alsotried ®tting with that constant term, but it gave identical results, sowe saw no reason to include an unphysical empirical term.

Table 1. The number of counts (N ) obtained from one of the spectra(i.e. 1 data-set), their precision ("N [%]) together with their ratios tothe reference values (R) and their precision ("R [%]). The right sideof eq. (2) contains rows 1---3, while either row 4 or row 5 is on theleft side of eq. (2). Simultaneous evaluation of 7 such data setsresults in an overall precision of 4---5% of the occupancy values

Ele N "N[%] R "R[%]

Fe 50280 0.7 1.40 1.0Al 10377 3.2 2.08 4.5Si 8706 3.1 0.776 4.4Mn 1545 7.2 1.51 10.1Ca 2491 4.4 1.38 6.2

Sites are Separable in Garnets with ALCHEMI 491

Conclusions

We predicted on a theoretical basis and proved by

quantitative experiments that all three crystallographic

sites can be distinguished in garnet samples by

ALCHEMI.

A key step in achieving separability and avoiding

unphysical values in the obtained occupation numbers

was the application of a formula with 3 independent

variables (internal standards) as contrasted to the

application of a formula with only 2 variables that had

previously been used in papers that were unable to

distinguish the X-site from the Z-site. Selection of 2

different natural garnets, in which the same elements

behave differently as far as substitution into the

different crystallographic sites is concerned, was also

a cornerstone of the experimental evidence. We

proved that Ca substitutes Fe and both sit on the X-

site in almandine, while in grossular, Fe and Ca sit

into different sites, because Fe substitute Al in the Y-

site while Ca retains its position on the X-site. So,

separation of both the X-sites and the Y-sites from the

Z-sites and from each other is unambiguous and the

obtained numerical values for the occupancy numbers

are in accordance with general mineralogical knowl-

edge.

Acknowledgement. Financial support by the Hungarian NationalScience Foundation (contract numbers OTKA 015895 and OTKA030432) is acknowledged. The authors are indebted to Dr. Stadel-mann for providing a copy of his EMS program.

References

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1543.[4] R. Balboni, S. Frabboni, R. Rinaldi, S. Spigarelli, Mikrochim.

Acta 1994, 114=115, 187.[5] M. T. Otten, P. R. Buseck, Ultramicroscopy 1987, 23, 151.[6] P. Stadelmann, Ultramicroscopy 1987, 21, 131.[7] C. J. Rossouw, C. T. Forwood, M. A. Gibson, P. R. Miller, Phil.

Mag. 1996, A74, 77.[8] J. L. LaÂbaÂr, Appl. Phys. Lett. 1999, 75, 70.

492 Sites are Separable in Garnets with ALCHEMI