M Ossbauer Absorption By Ions In The Highly Viscous Liquids

In the framework of hydrodynamic approach dynamical properties of ions are studied taking into account solvation. Density proole near the ion is described by the model potential which depends on the distance from the centre of ion. Amplitude of the potential is considered to be a uctuating quantity. Together with random location of ions these amplitude uctuations lead to additional disordering of a viscous liquid. For a solvated ion the linearized hydrodynamical problem is solved. For the model two-parameter potential, expressions for the dissipation function and diiusion coeecient are found. Obtained theoretical MM ossbauer spectrum has the asymptotics which is typical for highly viscous liquids.


Introduction
Spectra of resonance gamma-quantum absorption by the dilute electrolyte solution in highly viscous liquids 1] -4] contain a wide information about thermal motion of ion together with its environment.The simple analysis of the absorption spectra was done in the rst works on Sn 119 in polymer 2] and on Fe 57 in glycerol 3].Despite of the success in examining the dynamical properties of electrolytes 5,6] and highly viscous liquids 7,8], the rigorous analysis of the spectra of M ossbauer absorption by ions in supercooled liquids is still absent.The M ossbauer absorption measurements on ions of ferrous chloride in glycerol 1] demonstrate nonlinear dependence of broadening on the ratio of temperature to viscosity, T= .The dependence is close to the empiric Kohlrausch-Williams-Watts's dependence 9] -11] which is frequently used in processing the dielectric relaxation data, NMR experiments or light scattering by supercooled amorphous liquids, polymers, other disordered systems.
In our hydrodynamic approach the dynamical properties of ions are studied taking into account solvation e ect.The density pro le in the vicinity of an ion in the rest is described by the e ective potential (r i ) of the force acting on solvent molecules 12] (r i ) = 0 exp ?" i k B T (r i ) ; ( where 0 is an average density.This e ective potential is considered to be known and includes both the direct in uence of the ion and indirect in uence A.V.Zatovsky, A.V.Zvelindovsky through solvent molecules.As glycerol molecules are highly anisotropic and form into clusters at low temperatures 8], the nearest environment of localized ions is di erent.It can be taken into account by the random dimensionless parameter " i , which we take to obey the Gauss law with zero average value and dispersion h" 2  i i.
In our approach the problem is solved in the following way.Firstly, the hydrodynamic kinetic energy and the dissipative function of a separate moving ion are found at some value of " i .The obtained hydrodynamic elds allow us to write the time dependent velocity correlation function of an ion.This correlation function gives the spectral density of incoherent Van Hove function 13] that describes the M ossbauer absorption spectra.Then the results are averaged over distribution of " i .

Mobility of solvatons in solvent
Due to small compressibility of real liquids the absolute value of the ratio (r)=k B T is small.It is clear from equation ( 1) that the force acting on unit of liquid mass near the ion equals where p is the pressure, c is isothermal sound velocity.Let us consider the moving ion in the coordinate system with origin at the ion.Then the problem reduces to the rest ion in the stationary ow with the velocity u far from the ion.Due to the ow the density pro le in the vicinity of ion changes (r i ) = 0 exp f? (r) + (r)g ; ( where (r) = " i k B T (r i ) and is the correction term describing the change of solvate shell due to the incoming ow.Using equation (2) the hydrodynamic equations of motion of liquid near the ion can be written in the form div v = 0; (vr)v = ?c 2 r + v + 1 + 1 3 r div v; (4) where and are bulk and shear viscosities respectively.The elds v and must be regular everywhere, and v !u at r ! 1.We will search solution to (4) in the form v = u + r'.Then taking into account that u=c is small compared to unit one can nd, after linearization of equations of motion, 12] ' = u cos 1 r 2 Z r 0 r 2 (r)dr; (5) where cos = ru=(ru).
The dissipation in the uid is determined by the expression F = Z 2 (r i r k ') 2 + ? 2 3 ( ') 2 dV (7) and the total energy on motion of the uid is where integration is performing over total space.We choose the potential of an average force in the form of the twoparametric potential (r) = ?"0 tanh(r=r 0 ) r=r 0 2 ; (9) that is bounded at r = 0 and smoothly decreasing at r ! 1.Then the dissipation function and the energy (7,8) have the form F = 1  2 u 2 6 r 0 A 2 ~ ; A " 0 " i k B T ; ~ 1 3 + 1 + 2 ; (10) E = 1 2 u 2 MA 2 ; M 4 3 0 (r 0 L 2 4 1 + r 3 0 4 ) ; L + Now it is easy to nd in a usual way 14] the time dependent velocity correlation function of the solvated ion (solvaton) by means of equations (10,11) (t) hu(t)u(0)i = 3k B T M e ?t= 1 ; Note that the characteristic relaxation time 1 in equation ( 12) essentially depends on viscosity directly and via L.
Let us write also the di usion coe cient of solvation D 0 = 1

Cross-section of resonance gamma-quantum absorption
The e ective cross-section of the resonance gamma-quantum absorption is determined by the autocorrelation Van Hove function 15] g(!) = 1 N Z dt e ?i!t?jtj= N G s (q; t); (14) where h! is the di erence between energy of incident gamma-quantum and the resonance one, N is the lifetime of the exited state of nucleus, hq is the gamma-quantum impulse, and G s (q; t) = hexp(?iqr(0))exp(iqr(t))i ( 15) is the Fourier transform of the incoherent Van Hove function.The angular brackets denote the statistical averaging.There are several model approaches to nd Van Hove function.Most of them use the unfounded assumptions about time dependencies of the unknown functions and involve into consideration a lot of tting parameters.We use the fruitful approach 10,13] based on the description of the condensed media by the projection operator method.We choose the Fourier transforms of the uctuating local density of absorbers of gamma-quanta as the dynamical variables which are subject to averaging in equation ( 15) (see equations (12,13)).Here we use the approximation that corresponds to the commensurate relaxation times of the ion velocity and coordinate, 0 Note that in the limiting case of small velocity relaxation time compared to the coordinate relaxation time, 1 0 , the predominant term in equation ( 21) is I p ' 1 p + 1= 0 ; (22) which corresponds to the traditional approach.But after averaging over con gurations of " i we have

Concluding remarks
The obtained results determine the time dependance of the incoherent Van Hove function and are equivalent to the empiric Kohlrausch{Williams{ Watts's low mentioned in Introduction.The analysis carried out has shown that the condition 1 0 corresponds to the supercooled state of glycerol.The general result (21) describes the M ossbauer absorption spectrum (14) and after averaging over con gurations of " i it can be expressed as follows g(!) = Re I i!+ 1 N : (24) The comparison of the presented dependencies with the experimental data 1] on the gamma-quantum absorption by the ions of Fe in glycerol demonstrates a good agreement the experimental temperature behaviour of the spectrum parameters with one found theoretically.
Note that the method of the random potential of solvation is the hydrodynamical analog of the description of the various disordered systems such as, for instance, glasses.

For
of the method and approximations described in 13] we nd the integro-di erential equation dI

1 :
Using the results of the viscosity measurements at di erent temperatures 1] one can nd these relaxation times in a wide temperature range.Equation (18) can be solved by means of Laplace transformation pI p The result can be written in the form of continued fraction