Y. V. Chebrakov, V. V. Shmagin

St-Petersburg Technical University, 195251, St-Petersburg, Politehnicheskaya 29, Russia

e-mail: chebra@phdeg.hop.stu.neva.ru

We suppose for measurement experiments with heterogeneous objects that the data analysis model has form [1]

where y_n is n-th value of dependent variable; n =1, 2, … , N; F(A, X) is a given approximative function; W(D, X) is unknown approximative functi on, having finite value on the experiment realisation {X_n}; e_n is n-th value of random variable; g is a function, allowing to describe the way of dependent variable measurement. Thus, (1) differs f rom the classical data analysis model in the presence of the function W(D, X), by which, in particular, one may simulate an influence of the object heterogeneity on researching properties.

For the sake of simplicity we will assume further that the main estimation problem in measurement experiments with heterogeneous objects is to determine a form of the function W(D, X) and discuss only t wo simple cases, in which the problem on determining the form of function W of (1) is solvable.

Case 1. Let us suppose, that one may obtain the correct approximative function G(B, X) from an initial (postulated) function F(A, X) by means of its modification and G(B , X) differs from F(A, X) in an additive term W(D, X):

where W(D, X) is some continuous on interval [D ₁, D ₂ ] funct ion; D ₁ and D ₂ are respectively minimum and maximum values of the independent variable X on the exper iment realisation {X_n}. Let us demonstrate if the expression (2) is fulfilled, then sometimes one is able to determine the form of the function G(B, X) {or W(D, X)} by analysing procedures of prima ry processing of experimental arrays, by which researches eliminate the different operational defects of the realised experiments.

Indeed, in a set of measurement experiments with heterogeneous objects (for instance, in a set of the natural science experiments) an inconsequent parametric approach is often used in the procedures of primary processing of experimental arrays. This approach has following features [1]:

a) it is assumed {usually in the implicit form}, that the investigated experimental dependence {y_n, X_n} is described adequately by the function G(B, X) of (2);

b) a quantitative estimation of the values of an unknown approximative function W(D, X) of (2) on the experiment realisation {X_n} is made and it is supposed

where {y_n} is an experimental set of the dependent variable values; {W₀(X_n)} is a quantitative estimate of the values of the unknown approximative function W(D,&nbs p;X) of (2) on the experiment realisation {X_n}; {} is a set of the dependent variable values after primary processing of arrays {y_n}.

It is evident to determine the form of the function G(B, X) in the discussed case one should replace the inconsequent parametric approach by consequent parametric one, in which (ibid):

a) the approximative function F(A, X) is replaced on the function F(A, X) + W(D, X) {see (2)};

b)  the values of vector parameter A and D are estimated on the initial experimental array {y_n, X_n}.

E x a m p l e   1. As an illustrative example, in which the case 1 is realised, we consider some sort of physical and chemical experiments [2], in which the temperature dependences of magnet ic susceptibility c for some magnetically diluted systems are researched.

Up to date the representation of experimental dependences c (T) of magnetically diluted systems is made on diagrams 1/c –T. And, for many investigated weak magnetic matters, a set of breaks or significant deviations from the linear dependence, which is graphical form of well-known Curie – Weiss law

is observed on curves 1/c –T. In (4) c is the experimental magnitude of specific magnetic susceptibility; T is absolute temperature, K; C and q are parameters of Curie – Weiss law, and C_theor (the theoretical value of C) is determined by the following expression

where m_eff is an effective magnetic moment of a paramagnetic ion of metal Me, Bohr magneton; M is a molecular mass of diamagnetic solvent; M_Me and M_Mr ar e atomic masses of diluted (Me) and replaced (Mr) metals respectively; x is a concentration of metal (Me) in the analysing solid solution, mole fractions [3].

It is generally known, that flattening a curvilinear part of the dependence 1/c (T) is often successful by means of revealing temperature-independent term c ₀ of experimental dependence c (T). Discussion of the physical sense of the constant c ₀ for a set of metals, alloys and magnetic diluted systems one can find (ibid). As for processing methods and interpretations of experimental curves 1/c (T) with a set of breaks, up to date they are not yet scientifically established. In the discussed sort of experiments the presence of a set of breaks on diagrams 1/c –T is often connected with changing the type or the sign of exchange interactions between paramagnetic ions or the presence of phase transitions, and the analysis of magnetic behaviour of weak magnetic matters is usually made separately for low an d high temperature regions of the dependences 1/c (T) with using correlations (4) and (5).

It should be noted, that for many salts, as it is stated by analysing a lot of experimental dependences c (T) in [4], a set of breaks on diagrams 1/c –T passes into nothingness, when for fitting data arrays instead of Curie – Weiss law (4) one uses the following its modification

where c ₀ is a temperature-independent constant. Unfortunately, the idea about the connection of a set of breaks on diagrams 1/c –T with the presence of a temperature-independent magnetism in experimental dependences c (T) had no further expansion and dropped out eventually of researcher’s sight. To support B.Cabrera’s idea let us prove correctness of the following statement:

Proof. As generally known [3], in the experiments on researching the temperature dependences of magnetic susceptibility c of weak magnetic matters, the decomposition of measured values c on temperature-dependent c ₁ and temperature-independent c ₂ parts is realised by means of subtracting temperature-independent constant c _D from c , where c _D denotes a diamagnetic additive and for every investigating matter the value of this additive is calculated by special elaborated methods. However, if a structural formula of the matter is out of truth or the temperature-independent pa ramagnetism has a great magnitude, which can not be calculated in the frame of modern theory of magnetism (ibid), the foregoing approach does not lead to required results. Therefore, in mentioned cases instead of the temperature-dependent part, ob eyed Curie – Weiss law (4), the dependences

will be shown on diagrams 1/c –T, where c _PD = c ₀ – c ₂ and is a non-zero constant. Using mathematical analysis methods, it is easy to prove, that the curve (7) is continuous and smooth at all values T except for a value T_n = – q – C/c _PD, and character of its concavity depends on a sign of constant c _PD. Consequently, for the discussed experiments if c _PD ? 0 then some monotone concavo or convex curves will be shown on diagrams 1/c –T. To finish the proof it remains for us to add, that it is always possible to approximate any monotone curvilinear dependence, determined on a discrete set of {T_n}, by a set of lines with breaks within a given error, except for the single case when c _PD = 0.

Thus, in the discussed sort of experiments F(A, X) and W(D, X) of (2) are equal respectively C/(T + q ) and c ₀.

Case 2. In agreement with the determination of [1] for any local non-precision measurement experiment

and the vector parameter value of approximative function F(A, X) can be estimated correctly only on the array {X.

From (8) we obtain, if function W(D, X) is continuous on interval [D ₁, D  _;2 ], where D ₁ and D ₂ is respectively minimum and maximum values of the independent variable X on the experiment realisation , then

and, consequently, one may solve the problem of revealing the type of function W(D, X) by the data array {Y – g(F(A,&nbs p;X); X} and correlation (9).

E x a m p l e   2. As an illustrative example, in which the case 2 is realised, we consider experiments [5], in which the temperature dependences of magnetic susceptibility c for a set of polycrystalline systems are researched.

As it is stated in (ibid), polycrystalline solid solutions exist at vanadium concentrations 0.01 ? x ? 0.195 and 0.79 ? x ? 1. It is naturally to expect, that

a) at vanadium concentrations 0.79 ? x ? 1 the magnetic behaviour of systems on temperature is the same as the one of the magnetic–concentrated phase (vanadium oxide) or, in other words, experimental dependences c (T) are to have a sigma–shaped form and to be described by the equation [1]

which is the analogue of “closing function”, proposed in [6] for description of so-called “fuzzy phase transitions” {phase transitions with passing intensity, regulated by means of selection of independent variables values}. In (10) c is the experimental magnitude of specific magnetic susceptibility; T is absolute temperature, K; c ₁, a and T_c are the parameters of a sigma–shaped dependence (10); th is the hyperbolical tangent;

b) at vanadium concentrations 0.01 ? x ? 0.195 dependences c (T) should be rather hyperbolical type and therefore they must be well approximated by the modified Curie – Weiss law (6). Finding a set of breaks on diagrams 1/c –T for discussed systems in [5] is implicit confirmation of this prediction correctness {see above case 1}.

Figure 1. Dependences Dc  for systems :

It is evident, that correlations (6) and (10) describe different physical phenomena. Therefore, assuming independence (additivity) of these phenomena, we make a hypothesis, that, for system at all vanadium concentrations, the behaviour of  c (T) on the non-truncated experiment realisation {T_n} is adequately described by equation

contained 6 parameters.

One may estimate the fitting quality of equation (11) by dependences Dc = c –C/(T+q ), presented for different vanadium concentrations in figure 1 {continuous curves in plots are the approximative dependences Dc =c ₀+c ₁th (a(T + T_c) )}. Thus, for the discussed system F(A, X) and W(D, X) of (2) are equal respectively C/(T + q ) + c₀ and c₁ th ( a (T + T_c) ).

E x a m p l e   3. In [7] magnetic behaviour of systems was investigated for both mono- and polycrystalline samples. By graphical analysing the experimental diagrams c —T and 1/(c –c )—T the authors of [7] came to conclusions that

a) one may describe all found dependences c (T) by Curie – Weiss law c = C/(T + q );

b) if in magnetically diluted systems the iron concentrations have the same values then one is not able to distinguish polycrystalline samples from monocrystalline ones by means of analysin g dependences c (T).

In figure 2 (triangles and circles) we adduce the temperature dependences Dc =  c – C _? /(T + q  ^? ) – c_0? , where parameters estimates of model (1) are computed on truncated data array (c_n, T_n)* derived from initial one by deleting outliers. The continuous curves in figure 2 are fitt ing results of the dependences Dc by (10). From analysing curves in figure 2 we conclude that for the discussed solid solutions

a) for adequate fitting the data array (c_n, T_n) one may use the approximative model (2) with functions F(A, X) and W(D, X) desc ribed respectively by (6) and (10);

b) to distinguish polycrystalline samples from monocrystalline ones on found experimental dependences c(T) it is sufficient to draw the residual plots Dc(T) =   c - C ^? /(T + q  ^? ) - c_0? and compare each with other.

Figure 2. Dependences Dc (T) for systems :

x = 0,0027 (1), 0,0054 (2), 0,0055 (3), 0,060 (4), 0,060 (5), 0,0126 (6), 0,0148 (7);