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SIMULATION OF MÖSSBAUER RELAXATION
SPECTRA IN THE PRESENCE OF APPLIED
EXTERNAL MAGNETIC FIELDS
D. Niarchos, V. Petrouleas
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, suppliment au no 12, Tome 37, Ddcembre 1976, page C6-729
SIMULATION OF
M~~SSBAUER
RELAXATION SPECTRA
IN THE PRESENCE OF APPLIED
EXTERNAL MAGNETIC RIELDS
D. NIARCHOS and V. PETROULEAS
Physics Department, N. R. C. i< Democritos %, Athens, Greece
R6sum6. - Un programme de calcul des spectres Mossbauer, tcrit initialement pour le 57Fe en prksence d'effets de relaxation dans des matkriaux polycristallins ordonnks magnktiquement, a ktk ttendu au cas correspondant a l'application d'un champ magnktique extQieur perpendiculairement ou paralli?lement aux rayons X. Ce programme de calcul utilise le formalisme de l'opkrateur de Liouville avec les formes habituelles des Hamiltoniens nuclkaires et klectroniques en tenant compte de distorsions axiales et orthorhombiques lorsque les axes principaux du gradient de champ klec- trique ne cofncident pas avec ceux du champ cristallin. Un modkle de calcul est propost pour
S = 312 et avec des valeurs typiques des param2tres dkcrivant l'Hamiltonien de spin ; les rksultats sont obtenus avec un temps de calcul raisonnable pour deux composts paramagnktiques de Fe3+. Un ajustement satisfaisant des spectres expkrimentaux ne peut &re obtenu que par I'introduction d'effets de relaxation.
Abstract.
-
We have extended a program, written for the calculation of Mossbauer spectra of 57Fe in the presence of relaxation effects in magnetically ordered polycrystalline materials, to treat the case of an applied external magnetic field parallel or perpendicular to the gamma rays. The program uses the Liouville operator formalism with the usual nuclear spin Hamiltonian and an electronic spin Hamiltonian including axial and rhombic distortions in the general case of non coincident crystal field and e. f. g. principal axis systems. Model calculations with typical values of the spin Hamiltonian parameters for S = 312 were obtained within reasonable computer times. The calculation has been applied to two cases of Fe3+ S = 312 paramagnets. It was shown that satisfactory fits to the experimental spectra could be obtained only with the introduction of relaxation effects.1. Introduction.
-
The magnetic perturbation technique has been used extensively in Mossbauer studies of diamagnetic and paramagnetic substances. The resulting spectra in the general case of poly- crystalline absorbers contain (in principle) an infinite number of lines and may be used for the determination of fine and hyperfine structure parameters for the iron ion. The analysis of these spectra is not always straightforward. Apart from the simple case of dia- magnetic compounds, which has been accounted for in detail [l], the spectra of paramagnetic substances are generally handled in either the fast or the slow relaxa- tion limit 12, 31. Extension to intermediate electronic relaxation has been prohibited up to now by the long computing times required by the procedures in common use [4]. Quite recently however the relaxation cal- culation has been greatly shortened by the use of the Liouville operator formalism [5, 61 and appropriatecomputational procedures
17,
81. On the basis of these(*) Post doctoral fellow of the Greek National Research Foundation ; Present address : Chemical Biodynamics Lab, Lawrence Berkeley Laboratory, University of California,
Ca. 94720, U. S. A.
developments calculations have been performed in several 57Fe systems with remarkable succes in simu- lating spectra, that could not be fitted previously [7]. Applications, however, of the above methods to the cases of relaxation spectra in the presence of external magnetic fields have not appeared so far. A computer program, written by Grow for an Fe3+ S = 312 sys- tem [8], has been applied to the simulation of relaxa- tion spectra in the presence of spontaneous magnetic ordering [g]. We have modified and extended this pro- gram to simulate relaxation line shapes arising from paramagnetic S = 312 polycrystalline materials in the presence of an applied field oriented parallel or perpen- dicular to the gamma-ray direction.
In the following we give first a brief outline of the theory and then we apply the calculation to representa- tive cases of S = 312 paramagnets.
2. Theory.
-
2.1 HAMILTONIAN OF THE SYSTEM AND ELECTRONIC RELAXATION.-
The Hamiltonianemployed in the calculation is the following
C6-73 0 D. NIARCHOS AND V. PETROULEAS
where
+ 2 P H a . s (la)
HQ =
-
1512 3 I -
-
4+
I-
I (lb)and X,,,(t) describes the emission of y-rays from the nucleus. The D term in (la) described an axial crystal field-distortion of the electronic states while E accounts for any rhombic distortion. H , is the applied magnetic
field assumed 9 20 G, so that the hyperfine interaction
is approximately diagonal with respect to the electronic states. Within this approximation the electronic opera- tors in (lb) and (Ic) have been replaced by their expec- tation values. XQ describes the interaction of the nuclear quadrupole moment Q of the excited state of Fe57 ( I = 312) (XQ vanishes for the ground state) with the e. f. g. tensor. The primed axes represent the prin- cipal axes of the e. f. g. and in general do not coincide with the unprimed ones in (la). q and q are equal to the largest component, Vz,z,/e, and to the asymmetry,
V x y -
Vy'y', of the e. f. g., respectively. 3&,, des-
vz,z,
cribes the magnetic interaction of the nucleus with the electronic spin and the applied external field Aniso- tropic contributions to the magnetic hyperfine interac- tion are generally small for ~ e ~ + , S = 312 systems and are neglected here. The parameter H,, is therefore a
scalar and has an ionic value of
-
220 kG per spin 1.The expectation value of the spin varies in time as a consequence of random spin transitions induced by the dipolar term Gi,(t). The rate of this fluctuation is determined by the parameter C. All possible relaxation
mechanisms (spin-spin, spin-lattice and exchange interactions) have been replaced by an effective two spin dipolar interaction between equivalent atoms. A proper adjustement of the rate parameter C / r 3 and the polar angles (cp,, 0,) of the radius rector connecting the
two spins accounts approximately for all possible sources of relaxation.
The transition probability between eigenstates i
and f of (la) are calculated by means of the following formula
where the subscripts 1 and 2 refer to the two interacting ions. The matrix W i f accounts only for the most probable equienergetic transitions, i. e. those having equal initial and final energies between the two atoms. Due to this assumption n( f ) represents the population of the final state regardless of the atom under conside-
ration.
2.2 THE LINE SHAPE. - The line shape, I(o), of the Mossbauer spectrum in a stochastic treatment is expressed as follows [5, 6, 81
where
here, I, m, and I, m, are the quantum numbers of the ground and excited nuclear states respectively, X,,, is the operator which induces the emission of a y-ray from the Mossbauer nucleus, p = r / 2 - o where o is the frequency of the emitted y-ray
r
is the natural line width of the excited state and P, is the occupation probability of the electronic stateI
a>
which is one of the eigenstates of X,, (la). The matrix %(p) is the Liouville matrix of the form :where W is the transition probability matrix (2) and X: is the Liouville operator which corresponds to X, = X Q
+
K,,
(lb, c) [6]. U ( p ) is labeled by three indices, one stochastic, corresponding to the possible electronic levels, and the other two quantum mechani- cal corresponding to the excited and ground states of the emitter. The dimension of the Liouville matrix is therefore equal to (2 I,+
1) (2 I,+
1) (2 S+
1). For 5 7 ~ e and S = 312, %(p) has dimensions 32 X 32.The calculation of the spectral shape is considerably abbreviated by the use of the formula [7,8]
where M,, =
<
I, m,1
X,,,/
I,, m,>,
I is the unit matrix and V, I are the eigenvectors and eigenvalues of the Liouville matrix, which however does not contain the o term.The matrix elements of the radiation operator X,,, in (4) for pure M1 transition (L = 1, M = 0,
+
1) areproportional to the expression f10J.
SIMULATION OF MOSSBAUER RELAXATION SPECTRA C6-731
short as~follows. For a given direction of the applied field with respect to the crystal field (unprimed) axes (la) is diagonalized and the spin expectation values as well as the transition probabilities (2) are calculated. The applied field and the spin values are then rotated to the e. f. g. principal (primed) axes. The Liouville matrix (3) is in turn calculated through (lb, c) and (2) and diagonalized. The radiation matrix elements (5) are calculated by rotating the quantization (2') axis by
means of the ~ ( l ) matrix either to the direction of the applied field or transverse to it. In the latter case a summation of the spectral intensities is performed over two mutually perpendicular directions in the transverse plane. For polycrystalline absorbers the above calcula- tion is repeated for several field directions over the first octant of the unit sphere and the resulting spectra are added. A 4 X 4 summation was sufficient for the present calculations and required 18 min. time in a CDC 3300 computer.
The program gave identical results in the fast and the slow relaxation limit with a program in use in our lab utilizing a different approach for the calculation of the radiation matrix elements [12].
3. Application to S = 312 paramagnets.
-
The known examples of S = 312 paramagnets belong toeither of the following three classes :
(I) Fe (dithiocarbamato), halides [13], (11) Fe (dithio~alato)~ halides [l51 and (111) Fe (l-2-dithio- lene) 1151. Only the Grst class has been studied extensi- vely 1161. These compounds are generally characterized by either a large positive
D
2 10 K or a negative D-
2 K. In addition D has been found to be either parallel or transverse to the primary axis of the e. f. g. tensor. Of particular importance to the present discus- sion is the fact that most of the S = 312 compounds exhibit relaxation broadening or splitting of the Mossbauer lines.Theoretically calculated polycrystaline spectra with the above representative values and directions of D and for various values of the relaxation rate are given in figure 1 and 2. The other parameters used in the calculation are the followimg :
T = 4 . 2 K , H , = l O k G , H h f = 2 2 0 k G , e2 qQ/2 = 2.85 mm/s
,
q = 0.16 and
r
= 0.26 mm/s (FWHM).
FIG. 1.
-
Theoretical polycrystalline spectra in an applied field Ha = 10 kG perpendicular to the gamma rays for variousvalues of the relaxation constant, C (cm/s). The corresponding values of the parameters (see text) are : T = 4.2K, Hat = 220 kG,
C6-732 D. NIARCHOS AND V. PETROULEAS
FIG. 2.
-
AS in figure 1 except that D =-
2.5 K.cp, and 8, in the dipolar term (Id) were set equal to 00 and 90° respectively. Inspection of figures 1 and 2 shows that the spectra with intermediate relaxation rates exhibit unique characteristics and cannot be approximated by a fast or slow relaxation analysis. Also the spectra with different values and directions of D show distinct features, which may be used for classification.
In the following we apply the calculation to two specific compounds.
3.1 Fe(ethdtc),Cl.
-
This compound, which belongs to class (I), his been studied extensively [16,17] and is characterized byD-
-2.4 K and E= -0.24 K. In addition, D is directed along the minor axis of the e. f. g. (V;.,.). The polycrystalline spectrum at T= 3 K and H, = 8.8 kG transverse to the gamma rays is shown in figure 3a. Comparison with the spectra in figure 2b suggests a C value close to 2 cm/s. The theore- tical spectrum in figure 3a was calculated with the above values of the parameters and C = 3 cmls. The small deviations from the experimental spectrum may be attributed to the following approximations of the analysis. The dipolar angles cp, and 8, were fixed to 00 and 900 respectively. The static effect of the exchange interaction which is important in this compound wasneglected. Another source for the discrepancy may be a possible polarization of the absorber crystallites by the applied field. As a matter of fact an integration over a restricted range of angles gives better simulation to the spectrum.
3.2 [~e(dithioxalato)~I] (Ph4P),.
-
This com- pound is an example from the second class. No struc- tural and crystal field data are available for this compound. Stereochemical considerations support, however, a square pyramidal arrangement within the molecule [l41 similar to that found in Fe(dtc), halides. Representative spectra at 4.2 K and 1.42 K in applied fields up to 60 kG are shown in figure 3 (by c, d). Comparison with figures 1, 2 shows that D % 0 directed along the V,,,, axis. The theoretical curves in figures 3 (b, c, d) have been calculated by using D = 6 K, E = 0.0 K, H,, = 180 kG, = 0.25,r
= 0.26 mm/s and the relaxation rate values listed in the figure caption.the spectra of [Fe(dithioxalato),~] (Ph,P), and other compounds of class (11) in either the fast or the slow relaxation limit. In fact, the failure of the latter treat- ment led to the construction of the program described here. Another interesting result is that a certain set of parameters (including the relaxation rate) leads to a unique spectral shape. This is concluded by an exami- nation of the theoretical spectra in figures 1, 2 and other spectra, not shown here, with intermediate D values. This type of analysis may be therefore applied as an additional factor in establishing reliable values for the spin hamiltonian parameters.
We would like to acknowledge many useful discus- sions with Dr. A. Kostikas and Dr. A. Simopoulos.
FIG. 3. - (a) Polycrystalline spectrum of Fe(ethdtc)zCl at T = 3 K in a transverse applied field of 8.8 kG. Solid line was calculated theoretically, and corresponds to C = 3 cm/s D=-2.37K, D / / V , , , . , E=-0.24K, pd=O, B d = 9 0 , Hhf = 220 kG. (b to d) Polycrystalline spectra of [Fe(dithi- ~ x a l a t o ) ~ I ] (Ph4P)z in transverse applied fields ; (b) T = 4.2 K, Ha = 8.8 kG, (C) T = 4.2 K, Ha = 60 kG, (d) T = 1.42 K,
Ha = 8.8 kG. Solid lines were computed by using D = 6 K, D // Vz,z,, E = 0, pd = 00, Ba = 0 HM = 180 kG, = 0.25 ; - 6 -4 -2 0 2 4 6 the corresponding relaxation rate values are 0.4 cm/s in (6)
U E L ~ C I ?V (MM/S) and (d) and 3 cm/s in (c).
References
[l] COLLINS, R. L. and TRAVIS, J. C., in Mossb. eff. Mefhodol.
ed. by Gruverman, I. J. (Plenum Press, New York) vol. 3,1967, p. 123.
[2] LANG, G. and DALE, B. W., AERE Report No. 7478. [3] M W , E., GROVES, J. L., TUMOLILLO, T. A. and DEBRUN-
NER, P. G., Computer Phys. Comm. 5 (1973) 225. [4] BLUME, M. and TJON, A., Phys. Rev. 165 (1968) 446. [S] BLUME, M., Phys. Rev. 174 (1968) 352.
[6] CLAUSER, M. J. and BLUME, M., Phys. Rev. B 3 (1971) 583. 171 SHENOY, K. and DUNLAP, B., in Proc. of the Int. Conf. on
Mossb. Spectr. Gracow, Poland (1975).
[8] GROW, J. M., Ph. D. Thesis, Oregon State University (1975). We are indepted to Prof. H. H. Wickrnan for a copy of this thesis.
[g] GROW, J. M. and WICKMAN, H. H., AIP Con$ Proc. 24
(1974) 215.
[l01 Rose, H. J. and BRINK, D. M., Rev. Mod. Phys. 39 (1967) 306.
[l11 TINKHAM, M. Group fieory and Quantum Mechanics (MC Graw Hill) 1964, p. l1 l.
[l21 LANG, G., J. Chem. Soc. (A) (1971) 3246.
[l31 MARTIN, R. L., and WHITE, A. H., Inorg. Chem. 6 (1967) 712.
[l41 ~LTINGSRUD, D., Ph. D. Thesis, Iowa State University (1975).
1151 EPSTEIN, E. F. and BERNAL, I., Chem. Commun. (1970) 136. [l61 WICKMAN, H., TROZZOLO, A. M., WILLIAMS, H. J., HULL,
G. W. and MERRIT, F. R., Phys. Rev. 155 (1967) 563. [l71 GANGULI, P., MARATHE, V. R, and M m , S., Inorg.
