RECOVERY OF THE ENERGY ACTIVATION SPECTRUM

FROM THE DATA OF ACOUSTIC SPECTROSCOPY:

THE SOLUTION OF THE INVERSE PROBLEM

BY THE TIKHONOV REGULARIZATION METHOD

 

Yu.A. Semerenko

B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of the Ukraine, Kharkov, Ukraine

 

It has been known [1], that for real crystal with defects the damping decrement  and defect of the dynamic Young's modulus  are defined by statistical averaging for a Debye expressions [1] on the whole energy activation spectrum U of local structural imperfections (relaxators), which is characterized by the statistical distribution function P(U):

,     

were , t0 - the period of attacks, U - the activation energy, k – the Boltzmann constant, D0 - the elementary relaxators "power", Cr - the relaxators concentration.

It has been shown in Ref. [2] that the energy spectrum of acoustic relaxation in high-purity Fe single crystal of orientation <731> can be described by a quasi-Gaussian function . In this assumption the theoretical dependences well describe experimental temperature spectra  and  in the vicinity of a-peak (U0=0.037 eV, t0=2.4·10-11 s) which is observed at temperature »54 К (vibration frequency of the order of 88 kHz). However at the temperatures around 15 К it was observed some discrepancy of experimental and theoretical dependences. It is represented expedient to decide a inverse problem - recovering P(U) from experimental dependences  or  and carrying out the comparison of the P(U) with the "guessed" expression PG(U). If we shall replace the infinite top limit of integration in eq. by top border of a spectrum  then the inverse problem for spectral function PT(U) will be reduced to the solving of the Fredholm integral equation of the first kind relative to P(U). The decision of this inverse task by a Tikhonov regularizing method [3] has shown, that PG(U) largely coincides with the calculated spectral function PT(U), however PT(U) has a local peak at 0.015 eV, which can testify to existence one more relaxation resonance. If additional resonance is described by the meaning of t0 appropriate to a primary resonance, then the singularities like as peak and "step" will be observed on temperature dependences  and  in the region 14-16 K. In Ref. [2] these singularities are not found. It can be connected with essential broadening of the primary relaxation resonance. However, in previously published study [4] the absorption peak localized in the order of 17 К was detected on the pure iron single crystal samples of orientation <100>.

 

1.A.S. Nowick, and B.S. Berry, Anelastic Relaxation in Crystalline Solids, Academic Press, New York (1972); Atomizdat, Moscow (1975) [in Russian].

2.V.D. Natsik, P.P. Pal-Val, L.N. Pal-Val, and Yu.A. Semerenko // Fiz. Nizk. Temp. 26, 711 (2000) [Low Temp. Phys. 26, 522 (2000)].

3.A.N. Tikhonov, V.Ya. Arsenin The methods for solution of ill-posed problems [in Russian], Nauka, Moscow (1979).

4.P.P. Pal-Val, V.D. Natsik, S. Kadečková // Phil. Mag. A 56, 407 (1987).