Anomalies in the Mössbauer spectrum due to soft modes

Solid State Communications, Vol. 17, pp. 101 l—1012, 1975.

Pergamon Press.

Printed in Great Britain

ANOMALIES IN THE MOSSBAUER SPECTRUM DUE TO SOFT MODES Ch. Wissel Fachbereich Physik der Universität, D-355 Marburg, Germany (Received 13 April 1975 by E. Moliwo)

It is pointed out that definite theoretical predictionson the influence of a soft mode on the Mössbauer spectrum can not be made. Cusped-shape anomalies in the Debye—Waller factor and changes in the line shape may be possible.

IN THE LAST years Mossbauer spectroscopy has been used to investigate phase transitions. At structural phase transitions a dip in the Debye—Walier factor has been observed in several substances.”2 This result has been interpreted by the existence of a soft lattice mode w(K).3 In that case w(K) decreases in the vicinity of a special wave number K’. The corresponding weakening of the restoring force give rise to an increasing in the mean square displacement (r2) and therefore to a decreasing in the recoilless fraction. On the other hand experiments on substances with soft mode behaviour have been performed, which show no decreasing in the Debye—Waller factor.4 Therefore the question arises, what a serious theory can predict. The frequency distribution of the ~quanta is givenby a(w)

—‘

Jdr exp (it(~

—

Wi,)

— .~

t I + Q(t)

or

c~2(K)= w~+ XK2 w(K) = ~o + bK2.

}.

(3) (4)

So we end up with an integral t

K2 dK

~4+xic~

For a soft mode c~tends to zero approaching the transition temperature and the integrand becomes

In the quasi.harmonic approximation5 the lattice part ~ = e_i~t)e~>,can be written in terms of the lattice modes =

and the polycrystalline case I e(K) q 12 = ~q2le(K) 2. Usually one omits the time dependent part of equation (2) and takes le(K) 12 to be independent of K. For small K values an optical mode frequency for the isotropic case is given by

J (1)

Q(t)

Here we have taken the high temperature approximation for the phonon occupation number 1 1 e~’~—1 ~hc~,(K)

a constant. From that we conclude that the whole Brillouin zone contributes to used1’6 the integral. neither equations (3) nor (4) may be but Thus one has to

Vq2~T N6ir2m ~

know the whole dispersion relation and its temperature dependence. Further I e(K) I may generally not. be independent of Kand temperature. The corres-

x

ponding knowledges are generally not available neither from experiments nor theoretically. Therefore a dis-

I $ Ie(K) 12K2 w2(K)

[—

I + cos ~~.(I)t]dK

+ hi

2~

I e(K) 2K2 c~(K) Sifl ~~~(K)t dK —

)

(2)

cussion on the detailed temperature dependence of the Debye—Waller factor makes no sense •1.6 Qualitatively one can see at once that if w(K) decreases for

.

loll

1012

ANOMALIES IN THE MOSSBAUER SPECTRUM DUE TO SOFT MODES

Vol. 17, No.8

o(w)

constant so that any way the Debye—Waller factor can show only a casped-shape anomaly instead of a

\

~

2

o r 2r 3r FIG. 1. Frequency distribution for a single line. The parameter is ~0/r.The maximum is normalized to unity. The dashed line belongs to the original Lorentz curve, some K values approaching the transition temperature and if e(K) I does not change, the integrand and therefore the first integral in equation (2) increases no matter how the dispersion relation is. In special cases Ie(K)I which is not a unit vector, may be zero e.g. for the Ti ion in SrTiO3 at the 105 K transition. This means, that the MOssbauer atom does not high participate on theapproximation soft mode. Theinapplication of the temperature equation (2) can be doubtful if we have a low transition temperatare. The quasi harmonic approximation may be bad at least if the soft modes are strongly damped. Surely the anharmonicities restrict the magnitude of the mean square displacement to values smaller than the lattice

For the time dependent part of equation(2) one argues usually in the following way. The term cos varies over a time interval of the order of an inverse lattice frequency which is for an optical mode usually much smaller than F~’in equation (I) and therefore it oscillates out. But for the low lying part of a soft mode this may not be correct. In order to get a feeling for the influence of it we have used equation (4). We are aware of the fact that equation (4) may be applicable only for vanishing small K values if w0 goes down. This is surely the case if the K’ value mentioned above is zero and a coupling to the acoustic phonons occur. If K’ *0 this coupling plays no role. If we take I e(K)I to be independent of Kand insert equation (4) in the cos c~(K)tterm in equation (2) then the integration can be performed in a good approximation. Inserting the result in equation (1) we get for the line shape Fig. 1. We see that if~0comes in the order of magnitude of r the line becomes broader and is no longer of Lorentzian shape. This is possible for very small w~ and very large r values only. Abroadening7 of the line has been reported for NH.,(S04)2-12H2O. Summarizing one has to confess that the necessary information is not available to make a theoretical prediction on the influence of a soft mode on the MOssbauer spectrum. Anomalies of the Debye—Waller factor and changes in the line shape must not, but can, appear.

REFERENCES I.

RIGA~MONTIA. &PETRINI G.,Phys. Status Solidi4l, 591 (1970).

2.

5.

MONTANO P.A. et al.,Phys. Rev. B3, 858 (1971); YAGNIK C.M. et aL ,J. App!. Phys. 42,395(1971); CANNER J.P. et al.,J. App!. Phys. 42,4708 (1971); BHIDE V.G. & HEGDE M.S., Phys. Rev. BS, 3488 (1972); BHIDE V.G. & DURGE M.S., Solid State Commun. 10,401(1972); PEBLER J. & CALAGE Y., Phys. Status Solidi (a) 19,565 (1973); SULTANOV G. et al., Ferroelectrics 8, 539 (1974). MUZIKAR C., et al.,Phys. Status Solidi 3, K9 (1963). e.g. CLAUSER M.J., Phys. Rev. Bi, 357 (1970); and in SrTiO 3. BUNC R. & ZEKS B., SoftModes in Ferroelectrics and Antiferroelectrics. North-Holland, Amsterdam (1974).

6. 7.

BORSA F. & RJGAMONTI A.,Phys. Lett. 40A, 399 (1972). GLEASON T.G. & WALKER J.C., Phys. Rev. 188,893(1969).

3. 4.