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QUANTITATIVE MÖSSBAUER SPECTROSCOPY
R. Collins
To cite this version:
R. Collins. QUANTITATIVE MÖSSBAUER SPECTROSCOPY. Journal de Physique Colloques,
1979, 40 (C2), pp.C2-36-C2-38. �10.1051/jphyscol:1979211�. �jpa-00218479�
JOURNAL DE PHYSIQUE
Collogue CI, supplement au n° 3, Tome 40, mars 1979, page C2-36
QUANTITATIVE MOSSBAUER SPECTROSCOPY
R.L. Collins
Physios Dept. The University of Texas at Austin, TX., U.S.A.
Résumé.- L'analyse quantitative des mesures de spectroscopie Mossbauer, telles que l'obtention du rapport Fe3 +/Fe2 +, est rendue difficile par la différence des valeurs du déplacement quadratique moyen <x2> du noyau Mossbauer placé en des sites chimiquement distincts. Cet <x2> dépend de la température. On décrit ici une méthode permettant de mesurer l'intensité d'absorption correspondant à <x2> = 0 pour chaque site; ceci conduit bien à l'abondance atomique correcte. Ceci demande l'en- registrement de plusieurs spectres entre 10 K et 100 K.
Abstract.- Quantitative analysis of Mossbauer data, as in the measurement of Fe3 +/Fe2 ratio, is made difficult because of the differing mean square displacements <x2> of the Mossbauer nuclei at the chemically distinct sites. This <x2> is temperature dependent. A methodology is now described which permits the determination of the absorption area appropriate to <x2> = 0, for each site, which area measures correctly the atomic abundance. Several spectra at temperatures of 10 - 100 K are required.
Introduction.- Mossbauer spectroscopy has proved exceedingly useful for the identification of chemi- cal species which incorporate suitable nuclides.
That is, qualitative analysis. Other spectroscopies share with Mossbauer spectroscopy the problem of quantitative analysis, namely that the area under a given absorption peak does not relate directly to the abundance. For example, absorption spectra in the infrared, visible, and ultraviolet may give strong absorption for one substance and weak absorp- tion for another, when equal amounts are present.
Quantitative analysis is possible in such cases, by the construction of curves of absorbance vs. concen- tration for each species of interest. Interferences are common, and the factors change with temperatu- re. The methodology described below is intended to avoid the construction of calibration curves, which hence extends the utility to unknown substances having distinctive Mossbauer spectra. The methodolo-
gy readily gives the ratio of atomic abundances and with a single calibration to establish the instru- mental variables, can give absolute abundances.
Theory.- The area A under the Mossbauer spectrum for a given compound and for suitably thin absorbers is proportional to the product of abundance and the Debye-Waller factor f = exp(- <x2>/X2) / ) / , where
<x > is the mean square displacement of the nuclei along the gamma ray direction. <x2> depends on the zero-point motion and on temperature. It is readily shown that an Einstein model of the lattice gives 121
<x2> = k T° coth T0/T ( 1 )
c
where c is the force constant of the assumed harmo- nic potential in which the nuclei move. The absorp- tion area is
A = A exp (- <x2>/*2) (2)
where A measures abundance and <x2> reflects ther- o
mal and zero-point effects. The function of tempera- ture T = T coth T /T has an interesting interpre- tation. T is the temperature which would be required of a classical system, in order that the kinetic energy be equal to that fo the real (quantum mecha- nical) system at temperature T. This concept was originated by Mazo and Kirkwood /3/, who dubbed it
"kinetic temperature". For a parabolic potential, the virial theorem requires that the average kinetic energy equals the average potential energy. Each is separately equal to 1/2 kT per degree of freedom.
A consequence is that tf2 kT coth T /T
In A = In A 2 ° _ (3)
° 4 M k2 TQ 2 *2
where M is the reduced mass of 5 7Fe. Assuming this to be the actual mass, and inserting the other cons- tants, one expects for 57Fe
In A = la A - H-2064 T coth T /T (AS O T 2 O O W
O
Figure 1 is a computer simulation of (4). The lower curve for each T is the plot vs. T, and the upper curve is plotted vs. T . I n general, T is not known and must be derived from the data. Too large a va- lue of T leads to the plot being concave upwards, and too small a T leads to concave downwards,
o
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979211
Fig. I : Synthesized p l o t of log A vs. T f o r t h r e e d i f f e r e n t s i t e s , c h a r a c t e r i z e d by To = 25, 50, and 100, and f o r t h e same number of absorbing atoms.
Above e a c h curve i s t h e s t r a i g h t - l i n e p l o t v s .
%I
T = To c o t h To/T. Note t h a t t h e low temperature p o i n t s a r e d i s p l a c e d more t h a n t h e high-temperature ones. The dashed l i n e s r e p r e s e n t e x t r a p o l a t i o n of t h e s e s t r a i g h t - l i n e s t o t h e l e f t a x i s t o l o c a t e A.
f o r each s i t e .
To i s chosen t o g i v e t h e b e s t s t r a i g h t l i n e p l o t . Note t h a t t h e e x t r a p o l a t e d s t r a i g h t l i n e s (dashed c u r v e s ) i n t e r s e c t a t t h e same p o i n t , A = A
.
Hence, although one cannot reduce t h e mean square d i s p l a - cement t o z e r o , even a t a b s o l u t e z e r o , one can dedu- ce what t h e a b s o r p t i o n a r e a A. would have been were<x2> = 0 by e x t r a p o l a t i n g t o T* = 0.
Problems encountered i n t h e a p p l i c a t i o n of t h i s methodology i n c l u d e t h i c k a b s o r b e r s , phase chan- g e s , incomplete r e s o l u t i o n of peaks, and anharmoni- c i t y . The use of an E i n s t e i n model i n ( 1 ) may seem o v e r l y crude. A c t u a l l y , t h e r e i s seldom need f o r a more p r e c i s e model because t h e d i f f e r e n c e s a r e o n l y a few p e r c e n t o t t h e t o t a l motion / 4 / . Thick absor- b e r s can b e d e a l t w i t h u s i n g s t a n d a r d a n a l y t i c a l procedures / 5 / . Phase changes must be avoided, s i n c e T changes d i s c o n t i n u o u s l y . R e s o l u t i o n i s a c o n t i - nuing problem, and i s a n o t h e r r e a s o n f o r computer f i t t i n g . Anharmonicity can cause l a r g e problems, a s d e t a i l e d i n t h e next s e c t i o n . R e s t r i c t i o n of tempe- r a t u r e t o a low range i s h e l p f u l .
Experimental.- Experimental problems i n t h e opera- t i o n of a 2-stage helium DISPLEX r e f r i g e r a t o r have prevented t h e a c q u i s i t i o n of d a t a a p p l i c a b l e t o t h i s methodology. However, d a t a i s a v a i l a b l e i n t h e l i t e r a t u r e which p e r m i t s a check on t h e a p p l i c a b i l i - t y of t h e concept.
Hohenemser 161 measured c a r e f u l l y t h e absorp- t i o n of '19Sn f o r 8
-
Sn metal f o r a wide range o f temperature. His d a t a , where f E A/Ao, i s shown a st h e f i l l e d c i r c l e s of F i g u r e 2.
0
F i g . 2 : P l o t of l n f vs. T ( l e f t c u r v e ) , and a f o r - ced l i n e a r p l o t of l n f v s . To c o t h To/T ( r i g h t cur- v e ) . To = 90. M a t e r i a l i s
B -
Sn. Data i s by Hohene- n s e r / 6 / .A f o r c e f i t t o a l i n e a r p l o t v s . To c o t h T ~ / T g i v e s theopen c i r c l e s and To = 90. The f i t i s q u i t e poor.
However, by r e s t r i c t i n g t h e d a t a t o 100 K and below, a s i n F i g u r e 3 , t h e p l o t of I n f v s . To c o t h To/T becomes n i c e l y l i n e a r and p a s s e s reasonably c l o s e t o t h e o r i g i n .
Fig. 3 : Same d a t a for B-Sn,restricted t o temperatu- r e s below 100 K. l n f v s . To c o t h To/T i s c l o s e l y l l n e a r , f o r To = 31.
(To = 31) The fitted slope is -8.97 x which is only approximately the expected -15.5 x (from ( 3 ) ) .
Preliminary data from our lab on ferrocenium fluoroborate and ferrocene indicate that the mean square displacements differ markedly between these related compounds. In equimolar quantities, the area ratio drops from about 4 at 77 K to 1 0 at room temperature, with ferrocene giving the larger area.
This suggests that really large errors can be made by assuming that area alone measures abundance.
Conc1usion~- The conditions under which relative quantitative Miissbauer analysis is possible are (1) no phase changes should occur in the temperature range of interest, typically 12
-
100 K, and (2) the sdmples should be thin, to avoid saturation, or else corrections must be made for the effects of saturation. Absolute quantitative analysis becomes possible if the experimental variables are calibra- ted by use of a known absorber, such as an iron foil. It may be possible to perform quantitative analysis on compounds identifiable from their M%- sbauer spectra, from data taken at one temperature only, provided that the reduced mass M and also T have been previously established for that compound and provided that matrix effects prove relatively small. Further data will be needed to establish the usefulness of this concept.References
/ I / Goldanski, V.I. and Herber, R.H., Chemical Applications of the Gssbauer Effect (Academic Press, N.Y., 1968) 30.
/2/ Collins, R.L., Physics Letters
66A
(1978) 153-4./ 3 / Mazo, R.M. and Kirkwood, J.G., Proc. Nat'l.
Acad. Sci. (U.S.)
41
(1955) 204./ 4 / Collins, R.L. and Cosgrove, J.G., J. Inorg. nucl.
Chem.
2
(1976) 507-10./ 5 / Hafemeister, D.W. and Shera, E.B., Nucl. Instr.
and Methods
41
(1966) 133./ 6 / Hohenemser, C, Phys. Rev.
2
(1965) A 185.Acknowledgements.- Financial support by the Robert A. Welch Foundation is gratefully acknowledged. The computer fits of the 6
