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,
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,
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,
4.P.P. Pal-Val, V.D. Natsik,