The MÃ¶ssbauer Effect and Its Application in Chemistry

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Downloaded by UNIV OF CALIFORNIA SAN DIEGO on March 19, 2016 | http://pubs.acs.org Publication Date: June 1, 1967 | doi: 10.1021/ba-1967-0068.ch002

Structural Studies of Compounds and Alloys Using the Mössbauer Effect P. A . F L I N N

1

Carnegie Institute of Technology, Pittsburgh, Pa.

The Mössbauer effect, although not a substitute for other analytical methods such as x-ray diffraction, can be used to obtain several kinds of structural information about solids. In favorable cases, it is possible to obtain rather detailed information about the electronic configuration of atoms and the local symmetry of their sites by measuring the isomer shift and quadrupole splitting. If more than one valence state of a given atom is present, a semiquantitative determination of the amount of each kind is possible. In solid solutions, the amount of local or long range order can be estimated, and in certain defect structures the relation between the active atoms and the defects can be studied.

Crystal structure—i.e., the relative positions of the atoms in a crystal^ line solid, is usually determined by methods such as x-ray diffrac tion, neutron diffraction, and electron diffraction. One certainly cannot determine structure directly by the Môssbauer effect; however, there are special cases where it is difficult to get complete information by diffraction methods. In some cases, two atoms may have similar scattering powers and hence they are hard to distinguish by x-ray methods. One may have a partially ordered structure where there is some distribution of different kinds of atoms among different sites. In special cases, the Môssbauer effect can be a powerful tool to understand better what is happening, but it has not become a substitute for standard diffraction methods. One aspect of structure for which the Môssbauer effect becomes quite useful is the study of defect structures—cases where there are vacancies or impurity concentrations at low levels, or the early stages of phase trans1

Present address: Laboratoire de Physique du Solide, E.N.S.M.I.M., Parc de Saurupt, Nancy 54, France. 21 Herber; The Mössbauer Effect and Its Application in Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1967.

22

T H E MOSSBAUER E F F E C T A N D ITS A P P L I C A T I O N I N CHEMISTRY


formations, when the segregation of atoms is on a scale which is difficult to detect with diffraction methods. The Mossbauer effect can also be used to study surfaces. Here again, low energy electron diffraction can be used, but it is a difficult method to apply, and i n certain circumstances one gains some interesting information with the Mossbauer effect. I regard the question of electronic structure as part of the general question of the structure of solids. W e want to know not only where the nuclei are located relative to one another but what the electronic wave functions are like. This, I think, is the most important use of Mossbauer methods because it is an area i n which one can learn very little by x-ray diffraction. Some years ago Weiss and DeMarco tried to determine the electron distribution i n the transition metals (20). They arrived at seriously wrong answers (2) because of the inherent limitations of x-ray diffraction i n looking at outer electrons, whereas the isomer shift and the quadrupole splitting of the Mossbauer spectra can give rather direct information about the electronic wave functions. I would like to begin by discussing what is known as the greatest tin controversy. Isomer Shift 57

1 1 9

Next to F e , S n is the second "best" Mossbauer isotope, i n the sense that the γ-ray energy, Mossbauer lifetime, and parent lifetime are all satisfactory. T o use isomer shift measurements effectively, one must know the relative change i n nuclear radius (8R/R). In the case of S n , the correct value for this quantity was indefinite, both as to sign and magnitude. 119

Herber (4) and I arrived at the same conclusion about the sign of SR/R on quite different grounds. The nuclear theory was i n controversy: Belyakov, by one method arrived at the wrong answer ( 3 ) ; Uher and Sorensen, using a different method of approximation (19), arrived at the right answer, but of incorrect magnitude. Calculations of this sort involve small differences between large quantities and are not very reli able. It was possible to determine the sign of BR/R quite unambiguously, even before this new experiment, simply by considering the two allotropie forms of tin, both of which have long been well understood. Gray tin, which has the diamond structure, is a semiconductor, and the band structure has been calculated theoretically by Cohen and Bergstresser (5) and also by Bassani and L i u ( I ) . T i n also has a metallic form—white tin—with a somewhat more complicated crystal structure. The coordina tion is higher, and the interatomic distance is substantially larger. This, again, has been studied extensively by non-Môssbauer experiments such as the deHaas-vanAlphen effect and the Gantmacher effect (7, 9). Weisz has performed a detailed band structure calculation (21). F o r

Herber; The Mössbauer Effect and Its Application in Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1967.

2.

FLINN

23

Compounds and Alloys


each form of tin, then, the band structure is quite well-known, both from theory and experiment. The detailed band structures are complicated, but it is possible to give a rather simple qualitative description of them. One might expect (and it actually turns out that way) that the wave functions of the solid are primarily derived from the 5s and 5p electrons of the free atom. The bands are made up primarily of the 5s bonding and the 5s anti-bonding wave functions, and also the 5p bonding and the 5p anti-bonding. For gray tin (diamond structure) the atoms are close together with an interatomic distance of 2.8 A . This has the effect of raising the energy of the anti-bonding 5s levels somewhat above that of the 5p bonding states; therefore the occupied bands are built up from 5s bonding and the 5p , 5p , and 5p bonding electrons. Crudely speaking, there are three 5p electrons and one 5s electron per tin atom i n gray tin. x

y

z

Z

(5S )

(5s5p )

10

(4d )

3

SnCI \ 2

Mg Sn Sn(ct)

SnS0 jj

2

4

SnBr

2

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2

SnC 0 2

SnF K SnF 2

4

2

Sn P 0 1

6

2

2

7

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3

2

SnS

Sn(SQ4)

2

SnC H40 4

Sn (P0 ) 3

Sn0

2

4

Sn (A$0 ) 3

2

2

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2

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3

,K SnCI 2

4

SnO(a)

SnCI -5H 0 4

6

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Mfl Sn |(77«K) 2

-1

0

I

ISOMER SHIFT (mm./secj

Figure 1.

1

119

Isomer shifts for Sn in various environments, relative to a Mg Sn source at 77°K. (12) 2

In white tin, the nearest neighbor interatomic distance is increased to about 3 A . As expected, this lowers the energy of the 5s anti-bonding functions, and in a considerable portion of the Brillouin zone the 5s antibonding functions lie below the 5p like functions i n energy; hence, there are substantially more than one 5s electron and substantially less than three 5p electrons per atom. Judging from the Môssbauer data (one could get this presumably from the band calculations with enough effort),


24



14

2

6

the configuration is something like 5s 5p - . The opposite result is essentially inconceivable; there is nothing I can think of which would bring anti-bonding 5p electrons below bonding 5s electrons in energy, which would be necessary to get less than one 5s electron in the metallic form. Hence, it seems quite clear that using knowledge about the allo tropie forms of tin we can determine the sign of the change in electron density at the nucleus in going from gray tin to white tin. W e also need a calculation of the effect on the charge density at the nucleus arising from these changes i n the outer electron configuration. This was carried out by Waber, Lieberman, and Cramer (14) using a relativistic selfconsistent field calculation for tin for several different configurations. A somewhat similar calculation using a non-relativistic method to begin with and perturbation theory, was carried out by Ruby et al. with essen tially the same results (16). These calculations show that there is no serious disturbance of the Is, 2s, and 3s and 4s levels by changes in the number of 5s and 5p elec trons. The leading term in the change in charge density at the nucleus is simply proportional to the number of 5s electrons. There are two second-order terms, one term which depends on the square of the number of 5s electrons, which is simply a shielding effect of one 5s by the other, and a term involving the product of the number of 5s and 5p electrons which is the effect of the screening of the 5p on the 5s. There is essen tially no effect of the 5p electrons on anything but the 5s, and these two second-order terms have coefficients which are down by somewhat more than an order of magnitude from the leading term. To a good first approximation, the charge density at the nucleus depends simply on the number of 5s electrons, and one can make good quadratic corrections to this. The isomer shift measurements on the two forms of tin, coupled with the knowledge of the way the. wave functions behave, establish unambiguously the sign of SR/R. It is also easy to understand the pres sure experiment reported by Panyushkin and Voronov (15). They find that the charge density at the tin nucleus decreases with increasing pressure, but there is nothing anomalous about this. As pressure reduces the interatomic distance, the energy of the anti-bonding 5s orbitals is raised, producing a transfer of electrons from 5s character into 5p char acter as the energy of s levels is raised relative to ρ levels. This is exactly what one would expect. To get a fairly good number for the change in nuclear radius, we can consider a wider class of tin-containing materials. The results in Figure 1 are the work of Lees (12). Here we have taken arbitrarily as our origin, the source, the intermetallic compound M g S n , which provides a convenient cubic environment for tin. It gives a line which has natural 2


2.

FLINN

width and which, if cooled to 77°K., has a recoilless-free fraction of about 0.7 to 0.8, so that one gets a rather good Môssbauer effect. Compounds i n which tin is covalently bonded lie near the center of the diagram—gray tin and several organometallic compounds. Stannic compounds lie in a comparatively narrow group near —2 mm./sec, while stannous compounds lie i n a large band to the right of zero. One immediate question appears: w h y the wide range of isomer shifts for stannous compounds? The tin ions cannot be i n the ideal Sn (5s ) state. What is happening? A l l of these compounds have compli cated crystal structures, with tin located at a site which does not have inversion symmetry; there is an electric field acting at the site of the tin. This electric field produces matrix elements between the 5s and the 5p states of the tin and therefore a mixing of the two states. The ground state under these circumstances is not a pure 5s state, but a mixture of 5s and 5p. This is confirmed by studying the quadrupole splitting of 2+


25


2

I 0

I

I 0.4

I

I 0.8

I

I 1.2

I \ 1.6

1 I 2.0

I

I

I I

2.4

2.8

ELECTRIC QUADRUPOLE SPLITTING IN MM./SEC.

Figure 2. Relation between isomer shift and quadrupole splitting for stannous compounds (13)



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T H E MOSSBAUER E F F E C T A N D ITS APPLICATION I N CHEMISTRY

these compounds along with the isomer shift. Since the 5s wave function is spherically symmetric, it produces no electric field gradient at the nucleus. The 5p wave function, of course, is not spherically symmetric. It does give rise to a rather large field gradient at the nucleus, and to a good first approximation the quadrupole splitting observed is simply proportional to the amount of 5p character in the ground state wave function. A plot of the isomer shift vs. the quadrupole splitting should give a straight line, and in fact, does for most stannous compounds, as shown in Figure 2 (13). Noting this, one can then extrapolate this curve to zero quadrupole splitting, which would correspond to an ideal stan nous ion. That establishes the point on the isomer shift scale correspond ing to an ideal stannous ion, which, so far as I have been able to find, does not occur in nature. There apparently is no case in which the stan nous ion occurs at a point of cubic symmetry. 3

Having the point for a 5s5p configuration ( gray tin ) and the extrapo lated 5s point, one has a scale for coverting isomer shift to charge density at the nucleus. W e have established the value of 8R/R and we can look to see where other points would come. For example, we can look to see where an ideal stannic ion, which would have the configuration 4 d , no η = 5 electrons at all would be on the scale. W e find that actually there seems to be one stannic compound—potassium fluorostannate—which is approximately an ideal S n . The tin is surrounded by six fluorines, and it is not surprising that it loses all its valence electrons to the fluorine. W e find some other stannic compounds fairly close—the sulfate, the tetrafluoride, the oxide, and the chloride. The chlorostannate, for some reason, is much less ionic than the fluorostannate. One then sees that the organotin compounds, such as tetramethyl tin chloride, tetramethyl tin fluoride, tetraphenyl tin, lie slightly to the left of the ideal covalent tin configuration, which again is not surprising. W e know that tin can lose outer electrons relatively easily, whereas carbon does not, so it is not surprising that tin in contact with carbon would undergo what appears to be some transfer of charge to the carbon. The phenyl group, which is unsaturated, seems to soak up somewhat more charge than the methyl group. The relevance of the Mossbauer effect to chemistry is that now one can reinvestigate compounds such as the tin tetrahalides which were previously thought to be understood. There are molecular orbital calcu lations, quoted by Goldanskii, based on the assumption that the tin tetrahalides are primarily covalent in character. The results are misleading simply because the basic set of functions used is inappropriate. The great value of the Mossbauer technique is that once hR/R is established, one then has a rather severe test to apply to theoretical calculations concerning the structure of compounds. If 2

10

+4

x



2.

FLINN


27

the wave functions are correct, one w i l l obtain correct δ values for the charge density at the nucleus and from the ρ wave functions, correct values for the quadrupole splitting. In interpreting the quadrupole splitting, the electric field gradients seen by the nucleus can be either primarily internal or primarily external. In cases such as the stannous ions, where one has a considerable ρ char acter to the wave function of the stannous ion, the effect is primarily internal—i.e., the field gradient produced at the nucleus by its own electrons i n a non-symmetric wave function is large compared with that from external sources. W h e n one has spherically symmetric ions (—e.g., the ferric ion), one can see quadrupole splittings arising from the noncubic field of the surrounding ions i n the crystal. That effect is normally much smaller than the internal effect. If an internal quadrupole splitting is present, you need consider only the effect of the wave functions of the ion itself, at least as a reasonable first approximation. Structure of Solids First, some questions involving the crystal structure of alloys and intermetallic compounds. Figures 3, 4, and 5 are from the work of Shirane and others on the iron-rhodium alloys (17). The iron-rhodium alloy system is rather interesting. If rhodium is added to pure iron, the rhodium is soluble i n both the low temperature body-centered phase of iron and also the high temperature gamma phase, which is face-centered. Initially the transformation temperature is not changed drastically by adding rhodium; a body-centered solid solution exists at low tempera tures, and a face centered solid solution exists at elevated temperatures., When more than about 20% rhodium is present, the transformation from the high temperature phase to the low temperature phase is sufficiently sluggish so that one can preserve the high temperature phase, at least i n part, by drastic quenching. Also i n the low temperature phase—some where about 20 or 25% rhodium—long range order appears, and con tinues up to approximately a 50-50 mixture. A t 50-50 the ideal cesium chloride structure exists, with one kind of atom i n the center of the cube, the other kind at the corners. In alloys which contain less than the ideal equiatomic amounts of rhodium, the same structure exists, but now since there is not enough rhodium to go around, some of the rhodium sites are occupied with iron. One can study this quite nicely with the Môss bauer effect. Figure 3 shows the Môssbauer spectra for an alloy of 75% iron and 25% rhodium after two different heat treatments. Since absorption rather than transmission is plotted, these curves are right side up. The upper spectrum is taken from an alloy which was annealed i n the low tempera ture field (cesium chloride structure), and there are two six-line patterns



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I

-10

I -8

I

-6

I

-4

I

-2

I

0

I

2

I

4

I

6

I

8

I

10

Velocity (mm/sec) Figure 3. Mossbauer spectra for ordered and disordered samples of an alloy of 25% rhodium and 75% iron. The disordered sample was quenched and contains retained austenite (17) as indicated with intensities in the ratio 2:1. A t the 25% composition, half of the rhodium sites w i l l have to be occupied by irons; consequently, there w i l l be half as many irons on rhodium sites as there are irons on proper iron sites. One can immediately identify the weaker set of lines with the irons which are on rhodium sites. The splitting of this set of lines is somewhat larger than that of the other, meaning that the magnetic field at the iron nucleus is larger if the iron is on a rhodium site and therefore has all iron neighbors. This is not surprising. There is also a slight isomer shift between the two sets of patterns, meaning there is a slightly different charge density at the nucleus for the iron in the two cases. If the alloy is quenched so as to preserve some of the high tempera ture gamma phase which is not ferromagnetic, one observes a single peak at the center in addition to the six-line pattern from the disordered solid



2.

FLINN


29

solution. The transformation is such that no ordering in the alpha phase can occur and one obtains an almost random arrangement of the iron and rhodium atoms. The peaks are somewhat broader than in the ordered arrangement because one actually has a superposition of many six-Une patterns for iron with slightly different local environments. The hyperfine field and isomer shifts are slightly different at the different iron nuclei, depending on just how many rhodium and iron neighbors each has i n the random solution. It is also interesting to look at alloys in this system at and near the equiatomic composition. Figure 4 shows just two lines of the six-line pattern—the two on the far right. A t 50% rhodium every iron is com pletely surrounded by rhodium neighbors, and the lines are perfectly

8 2 Velocity (mrn/sec.)

Figure 4. Portion of the Môssbauer spectrum for iron—rhodium alloys containing 48 and 50% rhodium (17) sharp. A t 48% rhodium there are a few irons which are necessarily on rhodium sites, and their contributions to the pattern appear immediately as small peaks indicated by arrows. In this case, we see that the Môss bauer effect is very sensitive to small deviations from perfect order. Figure 5 shows the results for a moderately dilute alloy (10% rhodium). The main peak is from irons which have only iron neighbors; a smaller peak comes from those irons which, because of the statistical nature of the solid solution, have one rhodium neighbor.


30


Figure 6 illustrates a more complicated situation. The sample was a plain iron-carbon steel—an iron foil carburized to about 5 atomic % carbon and then quenched. One sees a rather complex pattern. There is a large central peak from some untransformed high temperature facecentered phase of iron containing carbon in solid solution, "retained austenite." There is a strong six-line pattern coming from "martensite," a distorted body-centered solid solution of carbon in iron. W e also see a


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The MÃ¶ssbauer Effect and Its Application in Chemistry

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