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M. Hirvonen, A. Jauho, T. Katila, J. Pohjonen, K. Riski

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Abstract. — The density matrix formalism is applied to the interpretation of Mossbauer spectra of single crystals of K3Fe(CN)« taken with polarized y-radiation to find the average electric hyper-fine Hamiltonian of the Mossbauer nucleus. Experimental results are given for 57Fe nuclei in the

monoclinic and orthorhombic polytypes. The hyperfine parameters and the orientations of the principal axes vary from one polytype to another. Both polytypes investigated show evidence for more than one inequivalent lattice site, although they are not energetically distinguishable, and an attempt is made in one case to resolve the sites. Some limitations and possible improvements of the accuracy are also discussed.

1. Introduction. — The low-spin ferric compound A ~f>

K3Fe(CN)6 provides a well-known example of cova- / \ / \

lency in the bonding between the ferric ion and the / \ / \ cyanide groups. The basic cell is monoclinic with two / \ / \ ferric ions [1] (Fig. 1), but X-ray investigations have / 9 \ / \

revealed polytypism due to different ways of stacking \iy \^JM^/ \_/ \

the monoclinic cells [2]. The most common polytypes \ ® r=r-~==j8L / \ \

are the regular monoclinic (1 M or M D 02) and the \ ®W^<rpf^J»\<B n o \

double orthorhombic (2 Or or MDO,) structure. \ ^ P ^ ^ S ^ IVy'b \

The magnetic properties of K3Fe(CN)6 have been \ h / ' \ "P

studied by several methods [3], but uncertainties \ / \ / remain which may be attributable to the presence of \ / ' \ /

different polytypes. It seems that the variation of the \ © /Af%^ Q< \ /

properties from one polytype to another has not been /ft&^ *r-~^^* \ / realized to a sufficient degree. A rather extensive /L$/^)P' \/

Mossbauer study of dilute and concentrated K3Fe(CN)6 /<y// °

has also been reported earlier [4]. ^<^y o Fe Recently, a systematic investigation of the spin o^ <B • C

structure and the hyperfine interactions below the ® N

magnetic ordering temperature (0.13 K) was

report-, TVreport-, TT. x- _*• / A * FlG- !• — The monoclinic unit cell of K3Fe(CN)6. The cell

eu p j . ine magnetic properties were touna to vary h a s

two inequivalent Fe3+ sites, one at the corner and the other

from one polytype to another. The investigation was at the center of the bc-face. The cyanide octahedra surrounding

carried out mainly by means of Mossbauer spectro- the Fe3+ sites are slightly elongated along axes lying

approxima-scopy with polarized •y-rays and single-crystal absor- tely m a plane normal to the c-axis (a'6-plane) and making angles bers, which is a powerful method of studying hyperfine °tf a b o u t 30° ™th th« 6"a™' ™ °^oaita, firections- T h e, f

. j . ,, , . . . . . atoms are not shown. The orthorhombic polytype is produced by interactions and especially their orientations m the alternate stacking of monoclinic unit cells and has four iron crystal also. By a properly chosen set of measurements sites.








Department of Technical Physics, Helsinki University of Technology 02150 Espoo 15, Finland

Résumé. — On applique le formalisme de la matrice densité à l'étude Môssbauer de mono-cristaux de K3Fe(CN>6 avec une source polarisée linéairement : l'interprétation des spectres permet de déterminer le Hamiltonien hyperfin électrique moyen du noyau Môssbauer. Les résultats expé-rimentaux sont donnés pour le noyau 57Fe dans les polytypes monoclinique et orthorhombique,

les paramètres hyperfins et les orientations des axes principaux variant selon les polytypes. Les deux polytypes étudiés présentent plusieurs sites de réseau inéquivalents mais énergétiquement indiscernables ; dans un cas, un essai a été fait pour les distinguer. Quelques limitations et des améliorations possibles de la précision sont aussi discutées.


C6-502 M. T. HIRVONEN, A. P. JAUHO, T. E. KATILA, 3. A. POHJONEN AND K. J. RISK1 the full hyperfine interaction operator for a given state

of the Mossbauer nucleus can be determined in a coordinate system fixed in the crystal structure. The Hamiltonian is conveniently obtained as an expansion in spherical tensors involving the nuclear spin opera- tors. If there are more than one inequivalent lattice sites, the method gives directly the average Hamilto- nian, but it may be possible to resolve the sites by a careful analysis. The purpose of the present investiga- tion was to determine the full quadrupole Hamiltonians at room temperature in the two common polytypes of K,Fe(CN)6 by Mossbauer measurements with plane polarized y-rays.

2. Sample preparation.


All the five samples used in this investigation were prepared from crystals grown at a constant temperature of


4 OC from a water solution on the bottom of a glass vessel. Under such conditions plate-like crystals with the a'-axis normal to the bottom were generally obtained. Sample 1 was made by selecting a crystal about 1 mm thick, casting it in epoxy, and thinning it down by grinding. Samples 2 and 3 were made from a common parent crystal by grinding slices cut in the desired directions. Samples 4 and 5 were prepared similarly from another parent crystal. The thicknesses of the samples were 0.3-0.4 mm. The crystal directions of the normals of the specimen platelets are indicated in table I.

The Mossbauer spectra of all the samples against a single-line source (CO in Rh) were taken below the magnetic ordering temperature. Samples 4 and 5 both gave a complex pattern (Fig. 2a), while samples 1, 2 and 3 gave well-resolved spectra having six lines at the same positions (Fig. 2b). The complex spectrum of sample 5 also contained lines at these positions. The different polytypes also gave different values of the quadrupole splitting at room temperature (see Table I). All the samples were also studied by X-ray methods. Diffraction spectra and Laue pictures confirmed that they were made up of the right compound, but failed to distinguish between the polytypes. However, pictures taken with a precession camera revealed the polytype structure (cf. Kohn and Townes [2]), and the results are given in table I. It should be kept in mind that the

FIG. 2.


Mossbauer absorption spectra of K3Fe(CN)6 single crystal samples below the magnetic ordering temperature against a CO in Rh source. a) Spectrum of the monoclinic sample 4 with the y-rays parallel to the 6-axis. 6) Spectrum ot the orthorhombic sample 2 with the y-rays parallel to the


precession picture is always taken of a very small test piece, and does not rule out mixed polytype structure. On the basis of the Mossbauer and X-ray precession data we conclude that the OD-structure of samples 1,2 and 3 is MDO,. Sample 4 we tentatively consider to be made up of the MDO, structure, while sample 5 does not show maximum degree of order.

3. Absorption matrix theory.


The density matrix of a nuclear sublevel ( $i


can be expanded in the

spherical tensor operators C; and Ss,

where I is the spin of the nuclear state [ 6 ] . The Hamilto- nian for the nuclear state is

where E, is the energy of the nuclear sublevel i. If there

Sample data

Crystal direction Mossbauer spectrum Quadrupole splitting Polytype indicated by Sample of plate normal below T , (mmls) at room temperature precession picture


MOSSBAUER STUDY OF ELECTRIC HYPERFINE INTERACTION IN K3Fe(CN),j C6-503 is no magnetic interaction, then for the 14.4 keV

excited state of 57Fe only second-order tensors enter into eq. (2), and the Hamiltonian is usually written in the principal axis representation as

The state of polarization of the y-radiation is repre- sented by a 2 X 2 density matrix a, in photon spin


A + B

A - B

where I)is the angle which the main axis of the polariza-

tion ellipse makes with the reference direction (Fig. 3).

FIG. 3. -The coordinate systems used in the analysis of the Mossbauer data. S is the source, A is the absorber, and SA is the direction of the y-rays being observed. SM is the direction of the magnetizing field in the source foil. X' y' z' is a rectangular coor- dinate system fixed in the absorber, and the y-ray direction is specified by the usual spherical coordinates 8', 9'. AM' is normal to SA and lies in the plane containing S A and the 2'-axis. It serves as the zero of the azimuth in the absorber. SM" is parallel

to AM'.

To each absorption line there corresponds a 2 X 2 absorption matrix in photon spin space,

whose elements depend on the direction of the radia- tion relative to the absorber, and on the expansion coefficients of the density matrices of the sublevels

(eq. (1)). For a given absorber transition due to a given

emission line the relative Mossbauer absorption in the limit of a thin absorber is proportional to

Hence, the elements of a, can be determined by observ- ing the intensities of the absorption lines with two different orientations of the absorber, and, generally, with two different source polarizations. Once the absorption matrices are known for each absofber transition, the expansion coefficients in eq. (1) can be obtained from their relations to the elements of a,

found in ref. [6].

4. Measurements and analysis.


The source consisted of about 10 mCi of 57Co diffused into a cir- cular area 6 mm in diameter in a 99.93


enriched 56Fe foil 25 pm thick. The foil was attached to a small button magnet mounted at the end of the vibrator rod. The y-rays were always observed in the direction of the normal of the foil, hence they were essentially plane polarized with E parallel to the magnetization M for lines 1, 3, 4 and 6, and E perpendicular to the magnetization for lines 2 and 5.

However, the foil is not magnetized to saturation, and the elements of a, for each line must be determined in separate experiments. Details of this procedure can be found in ref.


For the two polarization modes, E


M and E


M, we obtain (Tr c, is normalized to unity)

Figure 4 shows a typical spectrum. The twelve lines are numbered starting from the most negative velocity. In the fitting the lines were constrained to have equal widths and pairwise equal amplitudes (amplitude of line 1 = amplitude of line 11, etc.). Four independent intensities (lines 1, 2, 3 and 4) were used in the cal- culations.

FIG. 4. -A typical spectrum of a KsFe(CN)6 single crystal (two quadrupole absorption lines) against a magnetized CO in Fe source (six emission lines) at room temperature. The absorber is sample 2, and the direction of the y-rays is along the b-axis (8' = n/2, 9' = O), and y = 3 5 O . The intensities of the lines


C6-504 M. T. HlRVONEN, A. P. JAUHO, T. E. KATILA, J. A. POHJONEN AND K. J. RISK1 In table I1 we list the various orientations of the

y-beam with respect to the different samples for which we have data. We also give values of the angle



Orientations of the absorbers relative to the y-rays and values of


used in the measurements

Sample 8'







1 0 0 Fitted

7114 0




~ / 4 , 0

In table I11 we give the values of the elements of o, for the lower sublevel of the excited state, as determined from the various samples in the specified directions. For the upper sublevel the elements are obtained from those given in the table as 2-a, -c and -d, respectively.

Wherever possible, we have quoted the values resulting from fits to eq. (6). We have also normalized the values so as to make the a-elements of the two sublevels add up to 2 and the other elements up to 0. We note that using a non-polarized source only the elements a of a, can be calculated from the measurement data.

2 7112 0 Fitted TABLE 111

4 4 0 (),xi4 Elements of the absorption matrix of the lower

3 7112 7112 Fitted energy sublevel of the 14.4 keV state of 57Fe in 7114 ~ 1 2 0, n/4 K3Fe(CN)6 for various directions of the y-rays. Since

there is no magnetization in these samples, the elements

4 4 2 0 Fitted b = 0. The superstars indicate values obtainedfrom Jits

7114 0 0, n14 to eq. (6).

Fitted Direction O,n/4 of y-ray

8' 0' a C d Sample

indicate where we have fitted the $-dependence to eq. (6). Examples of the fitted curves are shown in figure 5. The fits also provide a check on the elements of the a,matrices in eq. (7).




FIG. 5. - The dependence on the angle y of the four indepen- dent line intensities in the Mossbauer spectrum of sample 2 with 0' = nl2, 9' = 0. The total intensity of the spectrum is normalized to 2. The curves are least squares fits to eq. (6).

The values obtained for the expansion coefficients of the density matrix of the lower sublevel are listed in table IV. These values are referred to the coordinate systems chosen for the analysis (Fig. 3). The transfor- mation to the principal axis system is made as explained in ref. [6], and the resulting values of the two nonzero

Expansion coeficients of the density matrix of the lower energy sublevel of the 14.4 keV state

of "Fe in K3Fe(CN),, as determilzed for various samples. The superstars indicateJitted data.

0 0

Sample CO C 2 2 0 1

Sum of


MOSSBAUER STUDY OF ELECTRIC HYPERFINE INTERACTION IN K3Fe(CN)a C6-505 coefficients c: and c; are also given in table IV. The

Hamiltonians obtained from eq. (2) are

in units of mm/s, and they are averages over the lattice sites. In the orthorhombic cases the main axis lies along the crystal d-axis within experimental error. In the monoclinic case the main axis lies in the

a' c-plane and makes an angle of about 30° with the a'-axis.

The sum of the squares of the second-order coeffi- cients (see Table IV) is less than one in each case, indicating the presence of different lattice sites. However, only one set of hyperfine energies is observed, hence the Hamiltonians at the different sites probably differ only by rotation of the principal axes, and the sites have identical values of the expansion coefficients in the principal axis systems. Since the average direc- tion of the main axis in the 2 Or polytype is a', we consider a model with two sites whose main axes are rotated through an angle o from the a'-axis in opposite directions in the same plane (cf. ref. [6]). The data on the first line of table I11 and the requirement ( ~ 4 ' -t (C:)' = 1 then give the results -c: = 0.988,


c2 = 0.057, o = 200, which correspond to a real Hamiltonian

referred to the principal axis systems.

5. Discussion.


In the calculations the experi- mental line amplitudes were used without thickness corrections. Since the line widths are equal, the amplitudes also represent partly the intensities of the lines, but saturation errors remain due to the effective thicknesses (about 1) of the samples. Uncertainties in the source calibration and in aligning the absorbers are other sources of error in addition to the statistical

error, which is relatively small. The errors can be somewhat reduced by fitting the $-dependence to eq. (6). The ratio of the elements c and d is practically unaffected by the thickness of the sample, while c2



is largely independent of alignment errors.

Although it is very difficult to carry out a complete error analysis, some methods of improving the accuracy can be enumerated. The application of a thickness correction to the fitted intensities and extreme care in aligning the samples are probably the most important measures to be taken. Accurate determination of the elements of the density matrices of the radiation is also essential, but the nature of the state of polarization is less significant. When different lattice sites are present an additional uncertainty may be introduced into the results by an anisotropic f-factor.

The Hamiltonians (8) demonstrate that the electric hyperfine interactions in the 2 Or and 1 M polytypes are different having different main EFG axes also. However, the accuracy of the determination was not high enough for reliable evaluation of v]. For example,

the Hamiltonians obtained for the orthorhombic samples, eqs. (Sa), (Sb), (8c), have somewhat different asymmetry parameters, and the principal X - and y-axes also turned out to be different. The direction of the main axis was found to be the same as below the magnetic transition temperature, namely along the a'-axis [5]. The low temperature measurements were done on the present sample 1, whose other axes also turned out to be the same as below T,.

We have demonstrated the interpretation of Moss- bauer polarimetric measurements with the aid of the density matrix representation by applying it to the determination of the electric hyperfine interactions at the iron nuclei in the 1 M and 2 Or polytypes of K,Fe(CN),. The two polytypes showed different hyperfine parameters and different orientations of the principal axes, and both of them gave evidence for more than one lattice site. Under such conditions the appa- rent value of q in the average Hamiltonian does not necessarily represent the actual asymmetry, and it is difficult to distinguish between the effects of real asymmetry and different lattice sites.



The authors are grateful to Prof. 5 . M. Daniels for useful discussions and to Outokumpu Oy for providing facilities for the X-ray investigations, and Mr. J. Yla-Jaaski for assistance in the measurements.


[l] FIGGIS, B. N., GERLOCH, M. and MASON, R., Proc. R. Soc.

A 309 (1969) 91.

[2] KOHN, J. A. and TOWNES, W. D., Acta Cryst. 14 (1961) 617 ; VANNERBERG, N.-G., Acta Chem. Scand. 26 (1972) 2863 ; For a general discussion ;of OD-structures see : DORN- BERGER-SHIFF, K., Lehrgang uber OD-Strukturen (Akademie-Verlag, Berlin) 1966.

131 GUHA. B. C.. Proc. R. Soc. A 206 (1951) 353 :



STEPHENSON, C. C. and MOR&, J. C., J. Am.: Chem.

Soc. 78 (1956) 275 ;

RISTAU, O., RUCKPAUL, J. and SCHOFFA, P., SOV. Phys.- JETP 8 (1959) 445.

[4] OOSTERHUIS, W. T. and LANG, G., Phys. Rev. 178 (1969) 439.





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