LATTICE DYNAMICS IN LAVES PHASES

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LATTICE DYNAMICS IN LAVES PHASES

S. Ramos-Bernal

To cite this version:

S. Ramos-Bernal. LATTICE DYNAMICS IN LAVES PHASES. Journal de Physique Colloques, 1979,

40 (C2), pp.C2-683-C2-685. �10.1051/jphyscol:19792237�. �jpa-00218619�

JOURNAL DE PHYSIQUE Colloque C2, supplkment au n o 3, Tome 40, mars 1979, page C2-683

LATTICE DYNAMICS I N LAVES 2HASES

S . Ramos-Bernal

Centro de Estudios Nuczeares, UNAM, Circuito Exterior, C. U., Mexico 20, D. F., Mezique

Rdsum6.- Le ddplacement total du spectre ~Essbauer des composds intermdtalliques ErFe' et ZrFe, a dtd mesur6 de 293 K l 700 K. Plusieurs hypothsses ont dtd faites pour extraire du ddplacement total la

contribution due aux propridt6s dynamiques du sous-ensemble fer. On a montrd qu'l haute tempdrature, le ddplacement thermique semble insensible aux ddtails des systsmes en vibration et qu'l trhs haute tempdrature il tend asymptotiquement vers la valeur classique. On a extrait les frdquences de Debye de ce sous-ensemble et trouvd qu'elles dtaient trop grandes. Des hypothhses suppldmentaires utilisant nos donndes de rayons X et le couplage entre les propridtds magndtiques et vibrationnelles du syst&ne semblent expliquer les dcarts observds.

Abstract.- We have measured the total shift of the Miissbauer spectra of the intermetallic compounds ErFe, and ZrFe2 from 293 K to 700 K. We have made several assumptions in the separation of the total shift in order to extract the dynamical properties of the iron sub-assembly. We have proved that at high temperature, the temperature shift seems to be intensitive to the details of the vibrational systems and at very high temperatures it approaches asymptotically to the classical value. We extrac- ted the so-called Debye frequencies for this sub-assembly and found to be too big. We made further assumptions using our X-ray data and the connection between the magnetic and vibrational properties of the system, which seems to account for the discrepancies that we first found.

Introduction.- When the center of a Mb'ssbauer spec- trum is shifted because of the difference in tempe- rature between source and absorber, we call it tem- perature shift. The decrease in the mass of the emit- ting nucleus affects its momentum and gives rise to a connection between the temperature shift and the dynamical properties of the solid /I/.

The fact that the magnetic exchange interac- tions depend on the distance between interacting ions suggests very strongly that the magnetic energy may influence the vibrational spectra. This influence, of course, will be proportional to the degree or or- dering of the spin systems, i.e. it should contribute differently in the ferromagnetic and paramagnetic regions.

the velocity of light, E the energy of the y ray and

< p >is the thermal average of the momentum.

T

Experimental.- The Miissbauer experiments were carried out with an arrangement suitable for transmission experiment using a 5 2 ~ o source which 14.4 rays were detected by a xenon filled proportional counter. The sample placed in a spacial furnace /3/ at 10-'torr and the temperature was measured using chrome1 alumel thermocouples. All samples which were in a power form did not show, after the high temperature nun, any sign of decomposition.

Dynamical properties.- In the case of a model for a solid distribution of frequencies we can give an expression, in the high temperature approximation, for the momentum and hence for the temperature shift:

The connection between the temperature shift

and the vibrational spectra hints at a possible link ₌

El3 IF

between this shift and the magnetic exchange inter- 2bi2c2 ZMC' ^8KT

-

exp hV

L\-1

action of the ions in the crystal lattice. 0 tKLI ⁽²⁾

It has actually been suggested / 2 / that the Unfortunately it is not possible to solve this temperature shift of the Miissbauer nucleus is affec- integral analytically, however, we obtained a stri- ted by the magnetization of the system. king similarity between our numerical calculations

Nonetheless, the existing models give an idea, and that using an expansion of the integral / 4 / in as a first approximation, of how to connect the dy- an infinite series in terms of the Bernoulli numbers.

namical properties of solids to the temperature shift It is well known that the Debye model is gross- which is given by ly inadequate for describing a solid, but the values

<p2 >

E = T

ZM'C' of the lattice properties and are merely an appro-

Where M is the mass of the emitting nucleus, c is ximation. Within this approximation, we have diffe-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19792237

C2-684 JOURNAL DE PHYSIQUE

rent Debye frequencies for the two kinds of ions in the crystal, however, they can be correlated to ob- tain the Debye frequency of the whole solid 151.

In particular, the well developed theory of an impurity 161 can be extended to polyatomic crystals.

Henceforward, we assume, as a first approxima- tion, that the subassembly of resonant ions In the crystal can be described by the Debye model.

For very high temperatures the classical limit of the temperature shift should be reached, however, experimentally this is frequently not the case. The- refore, let us find the deviation of the experimental data from the classical value.

The total experimental shift is given by adding the isomer and temperature shifts

Combining (2) and (3) we obtain

Where the left hand side is just the esperimental deviation from the classical value. Plotting the left hand side of equation (4) versus the reciprocal temperature, we should obtain a straight line whose slope is proportional to the square of the characte- ristic Debye frequency of the assembly of iron ions.

The values obtained for the.Debye frequencies for the compounds, do appear to be large compared with the values of metallic iron, and indeed too large for the expected values for a metal.

Let us improve the model by introducing some consideration that we have left aside so far, first, isomer shift is not temperature independent but it varies with temperature because of the expansion of the s wave functions. In the high temperatures region we can approximate it as linearly dependent on tempt?

rature, then we get;

where

i%)

pendent of'temperature, T is the temperature and a is a constant to be found.

In order to know the value of a it is necessary to have the following information 131 ;

i) The variation of the isomer shift with volume;

ii) The coefficient of thermal expansion.

Multiplying the values of i) and ii) we obtain the required value of a which differs for the ErFe, and ZrFe, -

.

We found aErFe = 0.3125 x

lo-'

m s-'

= 0.2125 x 10"

-

After taking into account the isomer shift variation, the equation (4) can be re-written as;

Another improvement to the theory is obtained taking into account that we are dealing with ferri- magnetic (ferromagnetic) materials, we should there-

fore consider the suggested /2/ effect that thechan- ge of the magnon spectra with temperature will affect the temperature change of the magnon spectra with temperature will affect the temperature shift. As we have said before, the change in the spin-ordering will influence the vibrational spectrum and this

should be reflected in the dynamical properties. Abo- ve the Curie temperature this effect, of course, should vanish.

Fig. 1 : Plot of the deviation of the data of ErFe, from the classical value vs. reciprocal temperature according to equation (6)

Fig. 2 : Plot of the deviation of the data of ZrFe, from the classical value vs. reciprocal temperature according to equation (6).

Using the method of Green functions /2/, Sh.

Sh. Bashkirov and Gr. Ya. Selyutin, found the effect that the magnetic energy has on a phonon state.

Applying their results to the Debye approximation References the expression for the temperature shift was found

to be;

with B = 8a2sis.a2 2 I" (a) V'M O

where 5 is the distance between neighbouring magne- tic ions, is the velocity of sound in the lattice, S is the spin of the ions and 1: is the second deri- vative of the exchange integral.

This result is just the same as that of equa- tion (2), but with a new temperature dependent Debye frequency. This temperature dependence is clarified using the molecular field approximation, where the temperature variation of the magnetization is the same as this new temperature variation of the Debye frequency, and can be expressed as;

v' = "(1 + Kopo 2 I ^{1 1 2} (8)

where KO is constant and U, is the reduced magneti- zation which we assumed to vary in the same way as the reduced effective field.

The "real" Debye frequency will be given by one obtained at the paramagnetic region, where

vo

i.e. the frequency is not affected by the magnetic ordering.

ErFe,

v

+

4

/I/ Josephson, B.D., Phys. Rev. Lett. (1960) 341.

/2/ Bashkirov, Sh. Sh. and Selyutin, G. Ya., Fiz.

Tverd. Tela

2

/3/ Ramos-Bernal, S., Ph. D. Thesis, Manchester, 1973.

/4/ Herberle, J., "Mb'ssbauer Effect Methodology", Vol. 7 (Plenum Press) 1971.

151 Isoilevskii, Ya. A., J.E.T.P.

7

161 Xaradudin, A . A , Montroll, E. and Weiss, J.,Solid State Phys. Suppl. 3 (Academic Press) 1963.