RU2054660C1 - Method for performing high precision x-ray spectrometry analysis of nonhomogeneous materials - Google Patents

Abstract

FIELD: material analysis. SUBSTANCE: method involves evaluating the contents of two C_A ingredients in two or more relationships (regressions) composed by using different groups of calibrated samples in the form of tablet-like samples differing from others taken from another group in factor levels inherent in the given group and invoking intensity degeneration of I_A line of the ingredient to be determined. A sample of material under study or material close in properties to that mentioned above is treated in a way which is not present in any of known methods to make an additional independent regression model called baseline regression. For the baseline regression an auxiliary regression model is built so that it has a common solution with the baseline regression model. Estimated A-component contents in a material is based according to the given method on the quantitative comparison of A-component line intensity in the material with those of A-component in mixed samples to be performed by applying calculation methods. EFFECT: enhanced accuracy of analysis. 18 dwg, 31 tbl

Description

 The invention relates to methods for x-ray spectral analysis (SAR) of raw materials and other materials and can be used in automated process control systems (ACS TP), quality control of raw materials and other materials in production and in laboratories where high demands are placed on accuracy and expressness analysis of inhomogeneous materials.

 A known method of SAR of non-homogeneous materials [1] is that at the same time fusion of the test sample and the comparison sample having known contents of the analyzed and “interfering” elements with flux is carried out, the intensities of the analytical lines of the elements being determined in the test sample and the comparison samples are measured, relations are determined the corresponding intensities, the coefficients of excitation, absorption and mass change during fusion, taking into account which determine the content of the analyzed elements.

 However, this method is not effective due to its low expressivity and accuracy.

(1) где α_A, α_Н параметры поглощения для определяемого элемента A и наполнителя Н соответственно;
К коэффициент пропорциональности, определяют содержание определяемого элемента в пробе путем сравнения интенсивностей определяемого элемента в анализируемой пробе и образцах-смесях.Closest to the invention is a method [2] partially removing the degeneration of the intensity of the analytical line of the element being determined, caused, in particular, by the influence of the dispersed (particle size) composition of the material. This method consists in the fact that samples P ₁ are obtained (made up) by adding certain amounts of diluent to the analyzed sample or samples P ₂ by adding specified quantities of the determined element to the determined sample, the intensities of the analytical line of the determined element in the analyzed sample and in samples P ₁ are measured or P ₂ and based on the initial type dependence
I _A =

(1) where α _A , α _H absorption parameters for the determined element A and filler H, respectively;
To the coefficient of proportionality, determine the content of the determined element in the sample by comparing the intensities of the determined element in the analyzed sample and sample mixtures.

 To compose expression (1), calibration samples are prepared from material close to the analyzed, and the line intensities of the determined element A in them are measured.

 Due to the lack of accuracy and rapidity, this method is not applicable under industrial control systems and for controlling the quality of raw materials in a production environment.

The aim of the invention is to significantly increase the expressivity and accuracy of the analysis of inhomogeneous materials without changing their aggregate state, in particular without using the effect of fusion of materials described in the analogue, and without compiling samples P ₁ or P ₂ and measuring the intensities of the analytical line of the element being determined in them according to the prototype.

 To ensure the expressivity of SAR materials in the developed method, computational operations and presentation (output) of output data should be carried out in an automated mode using a computer or computer.

 Graphical and analytical interpretations of the developed method of SAR materials are given in FIG. 1-18 and using the dependencies (1.1) - (4).

The essence and novelty of the developed method of SAR of inhomogeneous materials are as follows. In order to completely remove the degeneration of the intensity of the analytical line of the element being determined, caused, in particular, by the influence of the dispersed (granulometric) composition of the analyzed material, two or more dependencies (regressions) of type (1) are used. For their compilation, different groups of samples that are close in chemical composition to the analyzed material can be used, while the samples of one group must differ from the samples of another group, for example, disperse (particle size) composition. Let us call the indicated initial dependences basic. As another initial dependence of type (1), a theoretical dependence of the intensity I _{A of the} analytical line of the element being determined on its content C _A in auxiliary calculated samples can be used. We call the last initial dependence of I _A on C _A auxiliary. In this case, the basic and auxiliary dependences of I _A on C _A have a common solution with respect to C _A , i.e. their graphs intersect at one point.

(1.1) где
α

β

На фиг.1 базовая зависимость (1.1) графически представлена кривой 1. Рассмотрим точки А₁ и А₂, расположенные вне кривой 1.Consider the algorithms for compiling an auxiliary dependence I _A on C _A using the example of a basic dependence of type (1) given in the form
I _A =

(1.1) where
α

β

In Fig. 1, the basic dependence (1.1) is graphically represented by curve 1. Consider points A ₁ and A ₂ located outside curve 1.

(1.2) где C_A ω C_A1 + (1 ω) C_A2;
ω доля материала "образца" А₁ в образце-смеси, состоящей из материалов "образцов" А₁ и А₂.Any point outside curve 1 can be considered as an image of a “sample”, the material of which differs from the material of “samples” A in dispersion composition. For point A _{1, the} intensity of the analytical line of the element being determined with respect to the base point A ₀₁ with the same content of the element being determined is degenerate, for example, due to the difference in the dispersed composition of the material, for point A _{2, the} intensity of the analytical line of the element being determined with respect to the base point A ₀₂ the base curve is also degenerate. Let the materials for which points A ₁ and A ₂ are constructed, in composition (nomenclature) of the elements contained in them and the filler, coincide (or are close) with the materials of the base samples (points A ₀₁ and A ₀₂ ) and for simplicity the "interfering" elements do not contain . Then, with insignificant degenerations in the intensity of the analytical line of the element being determined in the “samples” A ₁ and A ₂ with respect to the basic “samples” A ₀₁ and A _02, we can write the following relationship, which is valid in the range (C _A1 -C _A2 ) of the contents of the element being determined ( assuming that the value of the parameter β with insignificant degenerations of intensity is practically independent of the dispersed composition of the material):
I _A = γ +

(1.2) where C _A ω C _A1 + (1 ω) C _A2 ;
ω the proportion of the material of the "sample" A ₁ in the sample mixture, consisting of the materials of the "samples" A ₁ and A ₂ .

и С_М:
α₁=

(1.3)
γ

(1.4) где ε_o tg Φ_o Φ_o оптимальное значение угла между кривыми зависимостей (1.1) и (1.2), положительное направление которого отсчитывается против часовой стрелки от кривой 1 базовой регрессии к кривой 2 вспомогательной регрессии кратчайшим путем.The parameters γ and α _{1 of the} dependence (1.2) are determined from the condition of intersection of the graphs of the base and auxiliary dependencies (1.1) and (1.2) at point M with coordinates

and C _M :
α ₁ =

(1.3)
γ

(1.4) where ε _o tan Φ _o Φ _{o is the} optimal value of the angle between the curves of dependencies (1.1) and (1.2), the positive direction of which is counted counterclockwise from curve 1 of the base regression to curve 2 of the auxiliary regression in the shortest way.

For basic regression (1.1), dependence (1.2) is auxiliary, the parameters α ₁ and γ of which, as can be seen from expressions (1.3) and (1.4), are quantities that depend on the abscissa С _{M of} point M.

The method for estimating the optimum value of ε _{o is} discussed below.

(2.2)
a₁= a+(b-b₁)C_М (2.3)
В разработанном способе РСА негомогенных материалов, как и в прототипе, определение содержания определяемого элемента заключается в аналитическом сравнении интенсивности I_AХ линии определяемого элемента в анализируемой пробе с интенсивностями определяемого элемента в образцах-смесях. Но в отличие от прототипа в разработанном способе сравнение интенсивности I_АХ c интенсивностями линии определяемого элемента в образцах-смесях производят с использованием двух или большего числа исходных (базовых и вспомогательных) зависимостей. Разработанный способ РСА обеспечивает полное снятие вырождения интенсивности линии определяемого элемента и, как следствие, достигается высокая точность анализа.If basic regression is given by expression
I _A a + bC _M , (2) the parameters a and b of which are determined from the base samples with known contents of the element being determined and its intensities, the auxiliary dependence has the form
I _A a ₁ + b ₁ C _M , (2.1)
where b ₁ =

(2.2)
a ₁ = a + (bb ₁ ) C _M (2.3)
In the developed PCA method of inhomogeneous materials, as in the prototype, the determination of the content of the element being determined consists in an analytical comparison of the intensity I _AX lines of the element being determined in the analyzed sample with the intensities of the element being determined in the sample mixtures. But unlike the prototype in the developed method, the comparison of the intensity of I _AX with the line intensities of the element being determined in the sample mixtures is performed using two or more initial (basic and auxiliary) dependencies. The developed SAR method provides a complete removal of the degeneracy of the line intensity of the element being determined and, as a result, high analysis accuracy is achieved.

(1.5) или C

=

(2.4) по следующим зависимостям:
I_A1= ω_oI_AX+(1-ω_o)

(1.6) или I'_A1 ω_oI_AХ + (1 ω_о)(a + bC_AБ), (2.5) образцах-смесях, состоящих из анализируемой пробы и вспомогательных образцов сравнения с содержанием С_АБ определяемого элемента, определяемым по выражению (1.5) или (2.4), по следующим зависимостям:
I_A2= ω_oI_AX+(1-ω_o)

(1.7) или
I'_A2 ω_oI_AХ + (1 ω_о)(a₁ + b₁С_AБ), (2.6) где ω_o оптимальное значение доли ω анализируемой пробы в образце-смеси.The expression of PCA in the developed method is achieved as follows. Using the basic and auxiliary regressions, the intensities of the analytical line of the determined element in the sample mixtures are determined by calculation: the sample mixture, consisting of the analyzed sample and the base comparison sample with the content of the determined element equal to
C _AB =

(1.5) or C

=

(2.4) according to the following dependencies:
I _A1 = ω _o I _AX + (1-ω _o )

(1.6) or I ' _A1 ω _o I _АХ + (1 ω _о ) (a + bC _AB ), (2.5) sample mixtures consisting of the analyzed sample and auxiliary comparison samples with the content of _{AB AB of the} element being determined, determined by the expression ( 1.5) or (2.4), according to the following dependencies:
I _A2 = ω _o I _AX + (1-ω _o )

(1.7) or
I ' _A2 ω _o I _АХ + (1 ω _о ) (a ₁ + b ₁ С _АБ ), (2.6) where ω _{o is the} optimal value of the fraction ω of the analyzed sample in the sample mixture.

b_Aj•C_Aj,
(2.7) где b_Aj коэффициент возбуждения (поглощения) интенсивности линии определяемого элемента А j-м "мешающим" элементом; A_j содержание j-го "мешающего" элемента в базовом образце сравнения.An example of justification of the optimal value of the parameter ω is given. It is assumed that the basic regression equation is obtained taking into account the influence of “interfering” elements contained in the base samples on the intensity of the analytical line of the element being determined and has the form
I _A = a + bC _A + bC _A

b _Aj • C _Aj ,
(2.7) where b _{Aj is} the excitation (absorption) coefficient of the line intensity of the determined element A by the jth “interfering” element; A _{j the} content of the j-th "interfering" element in the base comparison sample.

 Formula (2.7) is one of the forms of the Lucas-Tus method.

Then the intensity I _{A1 of the} analytical line of the element being determined in the mixture, consisting of the base reference sample and containing the “interfering” elements of the analyzed sample with a fraction of the latter equal to ω, can be expressed by the dependence
I _A1 ωI _AX + (1-ω) (a + b C _AB ) + (1-ω) f (b _Aj , C _Aj ). (2.8)
As can be seen from expression (2.8), for ω - >> 1 the last term of the expression can be neglected. Therefore, dependences (2.5) and (2.6), compiled for a mixture consisting of a basic comparison sample and an analyzed sample containing “interfering” elements, are valid for ω - >> 1.

If the basic regression is given by expression (2), then the intensity of the analytical line of the determined element in the mixture, consisting of the base comparison sample and the analyzed sample containing “interfering” elements, can be expressed by the dependence
I ' _A1 ωI _AX + (1-ω) (a + bC _AB ) (1-ω) f ₁ (b _Аj , С _Аjх ). (2.9)
It can be seen from expression (2.9) that, for ω - >> 1, the last term of the dependence can be neglected.

 The conclusions drawn about the optimal values of the parameter are also valid for dependences (1.6) and (1.7).

 Before considering the actual PCA method, based on the use of basic and auxiliary regressions, we consider a question related to terminology.

In SAR samples for which the intensity I _{AX of the} analytical line of the element being determined is degenerate relative to the base samples, there are two main cases of solving the problem. The case most often encountered in practice is when the direction of degeneracy of the intensity I _{AX of the} relative base samples is known. For example, the intensity of silicon in the 1KM sample (figure 2) is greater than the intensity of such a line of equal silicon content in the base sample of the K series (curve 1). We call the indicated direction of intensity degeneracy positive (I _AX I _AB >> 0), in the opposite case (I _AX I _AB <0) "negative", in the second case there is no information about the direction of degeneration of intensity I _AX relative to the base samples. In the first case, PCA to assess the content of C _{AX of} element A in the analyzed sample, it is enough to have one pair of basic and auxiliary regressions, in the second case, two pairs of basic and auxiliary regressions defined by dependencies (1.1) and (1.2) or (2) and (2.1) . Consider the SAR method of inhomogeneous materials, based on the use of two initial dependencies (basic and auxiliary regressions) for the case when the direction of degeneracy of intensity I _{AX is} known. Figure 1 shows a graphical illustration of the problem, from which it is seen that the degeneracy of the intensity I _{AX of the} element _being determined relative to the base regression is "negative" (I _AX I _AB <0). The _desired value of C _AX is located to the right of point C _AB .

+A

+A_3K= 0, (3) где А_1k ω_o(α _1k + βγ _k)(I_А1 I_А2k); (3.1)
А_2k=f( ω_o, β, γ _k, α_1k, С_AБ)·I_А1-f₁ (ω _o,β,γ_k,α_1k,C_АБ) ·IА_2k;
(3.2)
А_3k -(1- ω_о) γ_k C_АБ I_А2k, (3.3) где α_1k,γ _k параметры вспомогательной регрессии (1.2) согласно выражениям (1.3) и (1.4) для точки Мк c абсциссой С_Mк. На оси абсцисс существуют две практически симметричные относительно С_AБ точки С_Мk и C'_Мk, для которых справедлива зависимость (3).If the basic and auxiliary regressions are given by expressions (1.1) and (1.2), then to estimate the content of C _{AX of the} determined element in the analyzed sample, use the dependence
A

+ A

+ A _3K = 0, (3) where А _1k ω _o (α _1k + βγ _k ) (I _А1 I _А2k ); (3.1)
А _2k = f (ω _o , β, γ _k , α _1k , С _АБ ) · I _А1 -f ₁ (ω _o , β, γ _k , α _1k , С _АБ ) · IA _2k ;
(3.2)
A _3k - (1- ω _о ) γ _k C _AB I _A2k , (3.3) where α _1k , γ _k are the auxiliary regression parameters (1.2) according to expressions (1.3) and (1.4) for the point Mk with abscissa C _Mk . On the abscissa axis, there are two points C _Mk and C ' _Mk that are almost symmetrical with respect to _{AB AB} , for which dependence (3) is valid.

The ratio between the values of C _Mk and C _AX is expressed by the dependence
C _Mk C _AX + ψ where ψ is greater than or less than zero.

и С_Мk.As can be seen from expressions (3), (3.1), (3.2) and (3.3), dependence (3) contains two unknowns

and C _Mk .

To assess the content of C _{AX of the} element being determined using dependence (3), it is necessary to first evaluate the value of the quantity C _Mk . For this purpose, to the right of point C _AB on the abscissa axis, the interval l C _M1 - C _MN l is the content value of the determined element, in which the unknown content C _{AX of the} determined element should be searched. The interval lС _М1 - C _МN l is divided into equal sub-intervals with a step h, the aggregate of which forms a series of values of С _Мi :
C _M1 , C _M2 , C _MN . (3.4)
For each point of the series (3.4), they make up and solve the equation
A _1i C $\genfrac{}{}{0ex}{}{\text{2}}{\text{Ai}}$ + А _2i C _Аi + А _3i 0, (3.5) where С _{Аi is the} so-called current value of the content of the element being determined for the abscissa С _{Мi of the} point М _{i of the} intersection of the graphs of the base and auxiliary regressions;
A _1i , A _2i and A _{3i are the} coefficients calculated by the expressions (3.1), (3.2) and (3.3), respectively, for the abscissa C _Mi , for two values of the parameter ε _o (for example, at ε _o and ε _o ).

 As a result of solving equations (3.5) for series (3.4), we obtain the data given in Table 1.

(3.6)
В качестве параметра отличительного признака может быть использована величина K_М, равная
K_М=

(3.7)
Если параметр К_ε является универсальным критерием, то параметр К_Мобеспечивает необходимую разрешающую способность способа в тех случаях, когда вырождение интенсивности I_АХ линии определяемого элемента относительно невелико и базовые образцы и анализируемые пробы по своему минералогическому составу однородны.In order to estimate the value of С _Мk , for which dependence (3) is valid, use the method of optimizing the dependence of the current values of _Сi on the value of С _Mi using parameter K _{ε of the} distinguishing feature equal to
K _ε =

(3.6)
As a parameter of the distinguishing feature, the value K _M equal to
K _M =

(3.7)
If the parameter K _ε is a universal criterion, then the parameter K _M provides the necessary resolving power of the method in those cases when the degeneracy of the intensity I _{AX of the} line of the element being determined is relatively small and the base samples and the analyzed samples are homogeneous in their mineralogical composition.

A curve (table) is constructed (compiled) of the dependence of the parameter K _ε (or K _M ) on the values of the quantity C _{Mi of} series (3.4), the output curve (table) and the extremal value K _{ε is found.} The value of C _Mi , at which the parameter K _ε takes an extreme value, is an estimate of the quantity C _Mk .

cодержания определяемого элемента в анализируемой пробе в первом приближении. Для оценки содержания С_АХ определяемого элемента в пробе во втором приближении в качестве образца сравнения выбирают базовый образец с содержанием определяемого элемента, равным

, т.е. при C_АБ=

. В окрестности точки C_M=

на оси абсцисс выбирают интервал возможных значений содержания определяемого элемента, в котором находится искомая величина С_АХ. Строят (составляют) выходную кривую (таблицу), оценивают второе значение абсциссы С_Мk и по зависимости (3) оценивают

cодержания определяемого элемента во втором приближении. Если оценка

отличается от оценки

значительно, то производят оценку содержания определяемого элемента в третьем приближении, выбрав в качестве образцов сравнения базовый и вспомогательный образцы с содержанием определяемого элемента, равным

, и т.д. до получения практически постоянного значения оценки

.Solving the dependence (3) for the point M _k with abscissa With _Mk , we get the estimate

the content of the determined element in the analyzed sample in the first approximation. To assess the content of C _{AX of the} element being determined in the sample, in the second approximation, a base sample with the content of the element being determined equal to

, i.e. at C _AB =

. In a neighborhood of the point C _M =

on the abscissa axis, an interval of possible values of the content of the element being determined is selected, in which the sought value C _{AX is located} . The output curve (table) is built (compiled), the second value of the abscissa C _{Mk is} estimated, and according to dependence (3),

the content of the element being determined in the second approximation. If the score

different from grade

significantly, then evaluate the content of the element being determined in the third approximation, choosing as reference samples the base and auxiliary samples with the content of the element being determined equal to

, etc. until an almost constant estimate is obtained

.

C целью обеспечения экспрессности анализа вычислительные операции и выдачу выходных данных при оценке С_АХ cодержания определяемого элемента производят в автоматизированном режиме с помощью ЭВМ (компьютера).We note one feature of the search With _Mk in the case when C _Mi C _AB , which may take place in the second or subsequent approximations. In this case, I _A1 according to expression (1.6) and I _A2 according to expression (1.7) are equal to each other, I _A1 I _A2 and according to dependence (3.1) A _1i 0. The current value C _AI in this case is determined by the formula
C _Ai =

In order to ensure express analysis, computational operations and the output of output data when evaluating the C _AX content of the element being determined are carried out in an automated mode using a computer (computer).

+B

+B_3K= 0 (4)
где B_1k ω_obb₁ (I_А2k I_А1); (4.1)
B_2k f ( ω_o, a, b, а_1k, b₁, С_АБ) ·IА_2k f₁ x x ( ω_о, а, b, а_1k, b₁, С_АБ) ·I_А1; (4.2)
B_3k [а· а_1k + (1 ω_о) a_1k b C_АБ] ·I_А2k -[a ·a_1k + (1 ω_o) a b₁ C_АБ]· I_А1 (4.3)
Оценку С_АХ c помощью зависимости (4) производят по схеме, изложенной выше применительно к зависимости (3).If the basic and auxiliary regressions are given by expressions (2) and (2.1), then to estimate the content of C _{AX of the} determined element in the analyzed sample, use the dependence
B

+ B

+ B _3K = 0 (4)
where B _1k ω _o bb ₁ (I _A2k I _A1 ); (4.1)
B _2k f (ω _o , a, b, a _1k , b ₁ , C _AB ) · IA _2k f ₁ xx (ω _o , a, b, a _1k , b ₁ , C _AB ) · I _A1 ; (4.2)
B _3k [a · a _1k + (ω 1 _about) a _1k b C _AB] · I _A2k - [a · a _1k + (1 ω _o) ab ₁ C _AB] · I _A1 (4.3)
Evaluation of _AX using dependence (4) is carried out according to the scheme described above in relation to dependence (3).

Consider the work of the developed method of SAR of inhomogeneous materials. For this purpose, use experimental data on the effect of the dispersed composition of the analyzed samples on the intensity of the analytical lines of the determined elements. The starting material for obtaining samples of various disperse composition is silica SiO ₂ ChDA GOST 2428-60 (content of SiO ₂ 98.3% loss on ignition 1.5% residues 0; 2%). After grinding the starting silica in the mill, the final product was divided into two fractions: the first with particle sizes greater than 16 microns, the second less than 16 microns. When a small fraction is mixed with a large fraction in a certain proportion, an intermediate silica fraction is obtained. Further, these fractions are abbreviated as: small M, large K and intermediate KM. When polystyrene was used as a diluent from fractions M, K, and KM silica, samples with different silica contents were obtained. From these samples, “thick” samples were obtained by the pressing method for quantizing them to measure the intensity of the analytical line of the element being determined. The results of silicon quantization of these samples are given in table 2.

 A graphical illustration of the degeneration of the intensity of the analytical line of silicon in samples of one fraction relative to other samples according to table 2 is shown in figure 2.

As is known, dependences (3) and (4) are valid for optimal values of the parameter ε tgΦ i.e. at small angles Φ _o between the graphs (curves) of the base and auxiliary regressions. To justify the optimal value of the parameter ε, information on the intensity I _{AX of the} determined element in the analyzed sample can be used if there are no strict requirements for the expressness of the analysis.

Consider a specific example of the method of using a reference sample for preliminary substantiation of the value of ε _o For example, the raw material for ceramic products is a mixture of clay and perlite. The fraction of the latter in the mixture is a factor determining the degree of degeneracy of the intensities of the lines of silicon and aluminum. If the samples of the analyzed samples are arranged in a row in order of increasing percentage of perlite in them, then get
P ₁ , P ₂ , P _N , where sample P ₁ contains the smallest proportion of perlite.

к С_АХ. Ниже приводятся примеры решения указанной задачи применительно к зависимостям (3) и (4).The optimal parameter ε _o established for the sample P ₁ according to the intensity I _{A of the} determined element A in it, is optimal for all samples of the series. In this case, the sample P ₁ is a reference
The use of a reference sample is not limited to estimating the optimal value of the parameter ε. In those real SAR conditions, when stringent requirements are imposed on the expressness of analysis, it is necessary to determine in advance at what minimum number of approximations the goal is achieved. To this end, an assessment is made, to a first approximation, of the content of _{AX of the} element being determined in the reference sample at various fractions ω _{o of the} analyzed sample in the sample mixtures. With plotted _ACh from ω _o and determined value ω _'o, at which the maximum approximation

to C _AH . The following are examples of solutions to this problem as applied to dependencies (3) and (4).

 For example, 1, degeneration of the intensity of the analytical line of silicon in samples of the KM series relative to samples of the K series, we consider the justification of the optimal value of the parameter ε.

According to the silicon content and the intensities of its analytical line in three samples of the K series (Table 2), the values of the parameters α and β of the dependence (1.1) of the base regression are determined by the least square method and obtained (the intensity I _{AX of the} element being determined is expressed as 10 ^-2 I _AX )
α 4.8871; β = 0.01776.

The value of the parameter ω _о (the fraction of the analyzed sample in the sample mixtures) is taken equal to 0.95. Due to the absence of “interfering” elements in the K, KM, and K series, the use of dependence (3) is acceptable for all values of the parameter ω satisfying the condition 0 <ω <1. To justify the optimal value of the parameter ε _{о, the} dependence of the silicon content С _Аj is made (built) in the CM sample from the values of the parameter ε. These data are given in table.3.

Explanations for the table. 3. It is known that the degeneracy of the intensity I _{AX of} the silicon line in the 1KM sample relative to the base K samples is “positive”, and the interval of values C _M1 C _MN , in which the desired value C _AX is located, is to the left of point C _AB on the abscissa axis. According to the expression (1.5) With _AB 23.426607% As an auxiliary regression (1.2), we consider the variant shown in FIG. 1. In this case, the angle Φ is negative, therefore, ε _o has a negative sign. As the base and the auxiliary reference samples containing silicon samples c taken equal to 23.426607% C _AB In this case, I _A1 I _ACh. With _M taken equal to 22% as any point in the interval With _M1 With _MN on the abscissa. When calculating the parameters α _1i , γ _i and I _A2i , attention should be paid to the fractional part of the number: the larger the order of the fractional part ε _o , the greater the order of the fractional part (the number of significant digits) of these parameters. Condition I _A1 I _AX is fulfilled the better, the greater the order of the fractional part of the value of C _AB1 . According to the data in Table 3, for example, for ε -10 ^{-2 the} following roots C _{A1,2 get the} solutions of dependence (3,5): according to expressions (3.1), (3.2) and (3.3) with C _M 22% A ₁ 2 , 2947106 · 10 ^-2 , A ₂ 8.9474806 and A ₃ -208.68578, therefore, With _A1 22.073781% and With _A2 -411.99138%
Of the two roots given, choose a root satisfying min lC _M - C _A1 l0.073781% Therefore, the solution of dependence (3.5) under the above conditions is C _A 22.073781%
The dependence of the values of C _Aj obtained for the 1KM sample and the base regression with the parameters α 4.8871 and β 0.01776, on the values of the parameter ε is graphically shown in curve 1 of Fig. 3. From curve 1 of Fig. 3 it can be seen that over a certain range of ε values, the current values of C _{Aj of the} content of the element being determined are almost constant and with a further decrease in the values of the parameter ε, the values of C _Aj are subject to changes. The optimal value of the parameter ε is in the range where the values of C _Aj are subject to noticeable deviations (fluctuations). For the given curve I (figure 3), the optimal value of ε _o is in the range of l εl <10 ^-4 .

When l ε l ≥ lε _o l, the resolution of the mathematical model of the method decreases and practically drops to zero. In view of the heterogeneous materials described in the proposed SAR method, the parameter ε _o is the only characteristic experimentally estimated using a reference sample. Curve 1 of FIG. 3, constructed for the baseline regression sample system, is a characteristic curve.

Let us now consider the stage of evaluating the content of C _{AX of the} element being determined in the analyzed sample using dependence (3) using the example of the degeneracy of the intensity I _{AX of} the silicon line in the 1KM sample relative to the basic samples of series K.

Main (fixed) initial data to estimate the C content _{of ACh} in the silicon sample 1km given in Table 4.

It is known that the degeneracy of the intensity I _siх in samples of the KM series with respect to the basic samples of the K series has a positive sign. Consequently, the search for the content C of silicon _ACh 1km sample should be carried to the left of point C on the abscissa _AB1 where C _AB1 evaluation of silicon content in the sample 1km from the expression (1.5). When choosing the interval and the search ball h _{М Mk,} it should be borne in mind that their sizes are important for deciphering the output data of finding the extreme points of the output curve.

For the analyzed 1KM sample, the approximate values of the beginning and end of the interval and step h of the search for the desired value of C _Mk for silicon are selected (Table 5).

The data table. 4 and 5 are the initial data sufficient to estimate the content of C _{AX of} silicon in the 1KM sample in a first approximation.

Table 6 shows the results of a search for the C _Мk value for the silicon content in the 1KM sample to a first approximation. According to the data in Table 6, an output curve is plotted as shown in FIG. 5. When computerizing calculations and plotting, the output curve according to the data in Table 6 is displayed on the display screen.

20,78% (cм. строку 13 табл.6).Figure 5 shows that the dependence of the parameter K _M on the value C _M is a wave-like curve, the amplitude of which changes in a certain way: from the beginning of the search to point A, the amplitude of the output curve decreases, in section A-B the value of the parameter K _M practically does not change, and to the left of point B the magnitude of the values of the parameter K _M in the direction of the search increases. The extreme region of the output curve is plot AB. The point μ (the middle point of the AB section) is taken as the extreme point of the output curve. The projection of the point μ onto the abscissa is the desired value With _Mk . For the silicon content in the 1KM sample, as can be seen from Fig. 5, C _Mk 20.6% Solving equation (3) with C _Mk 20.6% and the initial data given in Tables 4 and 5, an estimate of the silicon content in the sample is obtained 1KM in a first approximation:

20.78% (see line 13 of Table 6).

, т.е.In the table. 7 and Fig. 6 show the output of the search for the value of C _Mk2 for the silicon content in the 1KM sample in the second approximation. In this case, samples (basic and auxiliary) with a silicon content in them equal to

, i.e.

= 20,8%
Выбор интервала поиска С_Мk осуществляют с учетом того, что искомая величина С_Мk2 может находиться на оси абсцисс как слева, так и справа от точки С_АБ2, а шаг поиска h₂ < h₁.C _AB2 =

= 20.8%
The selection of the search interval With _{Mk is} carried out taking into account the fact that the desired value With _Mk2 can be located on the abscissa axis both to the left and to the right of point C _AB2 , and the search step is h ₂ <h ₁ .

The search data in the third approximation of the С _Мk value of dependence (3) for the silicon content in the 1KM sample are given in Table 8.

20,19%
Результаты оценки С_АХ в третьем приближении получены на основании данных табл.8 и выходной кривой, приведенной на фиг.7:
C_Мk3= 20,175%

= 20,16%
По результатам оценок содержания кремния в пробе 1КМ получают

=

= 20,16%
Относительная ошибка РСА для кремния в пробе 1КМ составляет (С_АХ 20,18% табл.2):
σ_o=

• 100% -0,1%
Отмечают следующую закономерность разработанного способа РСА негомогенных материалов. Результаты оценок содержания определяемого элемента в первом и последующих приближениях зависят от доли ω_оанализируемой пробы в образцах-смесях. Указанная закономерность для реперного образца 1КМ и базовой регрессии с параметрами α 4,8871 и β 0,01776 приведена графически на фиг.4, из которой следует, что с увеличением значения ω_о эффективность оценки содержания определяемого элемента в пробе возрастает, так как при этом

__→ C_AX Ввиду изложе ного построения кривой зависимости

от доли ω_о анализируемой пробы в образцах-смесях (далее указанную кривую называют долевой) для системы реперный образец базовая регрессия должны предшествовать этапу непосредственного РСА анализируемых проб.From the output curve of the second approximation (Fig.6) it is seen that the desired value of C _{Mk2 is} 20.15% Solving equation (3) with C _{Mk2 of} 20.15% and With _{AB2 of} 20.8%, we obtain an estimate of the silicon content in the 1KM sample in the second approximation :

20.19%
The results of the assessment of _AX in the third approximation are obtained on the basis of the data in Table 8 and the output curve shown in Fig. 7:
C _MK3 = 20.175%

= 20.16%
According to the results of assessments of the silicon content in the sample 1KM receive

=

= 20.16%
The relative PCA error for silicon in the 1KM sample is (C _AX 20.18% of Table 2):
σ _o =

• 100% -0.1%
The following regularity of the developed SAR method of inhomogeneous materials is noted. The results of assessments of the content of the element being determined in the first and subsequent approximations depend on the fraction ω _{of the} analyzed sample in the sample mixtures. The indicated regularity for the 1KM reference sample and basic regression with parameters α 4.8871 and β 0.01776 is shown graphically in Fig. 4, from which it follows that with an increase in the value of ω _{о, the} efficiency of estimating the content of the element being determined in the sample increases, since

__ → C _AX In view of the above construction of the dependence curve

of the fraction ω _{of the} analyzed sample in the mixture samples (hereinafter referred to as the shared curve) for the reference sample system, the basic regression should precede the stage of direct SAR of the analyzed samples.

Example 2 SAR samples for which the intensity of the analytical line of the element being determined is degenerate relative to the base samples. Sample 1KM in relation to sample 1M (see table 2) is a reference sample. It is known that the degree of degeneracy of the intensity of the analytical line of silicon in the 1M sample is greater than in the 1KM sample. The search for the silicon content in the 1M sample should be searched to the left of the point _AB , estimated using basic regression. In the table. Figures 9 and 10 show the initial data for estimating the silicon content in the 1M sample as a first approximation.

≈ 33% (решение уравнения (3) при С_Мk 31,75%).A fragment of the output curve for estimating the silicon content in the 1M sample is shown to a first approximation in Fig. 8 (C _Mk1 31.75%). The result of the first approximation is

≈ 33% (solution of equation (3) at С _Мk 31.75%).

 In the table. Figure 11 shows the variable initial data for estimating the silicon content in the 1M sample in the second approximation.

32,8% (решение уравнения (3) при С_Мk 32,775%).From the output curve of the second approximation (Fig. 9) it can be seen that With _Mk2 32.775%, the Estimation of the silicon content in the 1M sample in the second approximation is

32.8% (solution of equation (3) with C _Mk 32.775%).

The search for the value With _Mk in the third approximation is performed using the parameter K _{ε of the} distinguishing feature (parameter K _M does not provide sufficient resolution of the method). The initial data (variables) for searching With _{Мk3 are} presented in Table ₁₂ .

The output data for the assessment of With _{MK3 are} presented in table.13.

32,78% (решение уравнения (3) при С_Мk3 32,85%). Следовательно,

=

32,78% Результаты оценок содержания кремния в пробах 2М и 3М с помощью базовой регрессии с параметрами α 4,8871 и β 0,01776: 22,67 и 26,41% соответственно.From the output curve of the third approximation (Fig. 10) it can be seen that With _Mk3 32.85% The results of evaluating the silicon content in the 1M sample in the third approximation are

32.78% (solution of equation (3) with C _Mk3 32.85%). Hence,

=

32.78% Results of assessments of the silicon content in samples 2M and 3M using basic regression with parameters α 4.8871 and β 0.01776: 22.67 and 26.41%, respectively.

 Of fundamental importance are the results of estimates of the silicon content in the K series samples, if we consider the M series samples as the basic samples. The following parameters of the base regression of the type (1.1 ): α 5.8388 and β 0.019483. The characteristic and beat curves for the indicated base regression and the 1KM sample practically do not differ from those curves shown in Figs. 3 and 4.

 The following positive results of assessments of the silicon content in samples 2KM and 2K indicate the versatility and high resolution of the developed method of SAR materials.

The results of estimates of the silicon content in the 2KM sample using basic regression with parameters α 5.8388 and β0.019483 at ω _o 0.92 and ε _o ± 8 · 10 ^-5 are given in table.14.

The results of estimates of the silicon content in sample 2K using basic regression with parameters α = 5.8388 and β = 0.019483 at ω _о 0.96 and ε _о ± 8 · 10 ^-5 are given in Table 15
In the above examples of SAR of inhomogeneous materials, there is a degeneration of the intensity of the analytical line of the element being determined, caused by the difference in the dispersed composition of the material of the analyzed sample from that of the material of the base samples. Of fundamental importance is the question of the applicability of the developed SAR method for sample analysis, the intensity of the analytical line of the element being determined is degenerate relative to the base samples due to the difference in the mineralogical composition of the material of the latter from the mineralogical composition of the analyzed sample. To solve this problem, data is used that contains information on the intensities of the analytical lines of the elements being determined in two groups of samples that differ in the mineralogical composition of their materials. In particular, Table 16 shows the data on the degeneracy of the intensities of the analytical line of silicon in fused clay samples relative to extruded clay samples. Moreover, in contrast to pressed clay samples, fused samples have the addition of Ingichkin quartz (KI). Using pressed clay samples and fused clay samples without KI additives, the following two regression equations of type I _si f (a, b, С _si ) were obtained:
I _si -0.1484 + 0.18322 С _si (5.1) for fused samples and
I _si -0.35573 + 0.0220264 C _si (5.2) for pressed samples.

 The data of column 5 in comparison with the data of column 3 (Table 16) indicate that the presence of (addition) of quartz in clay samples causes a degeneration of the intensity of the silicon line with respect to melted clay samples without CI additives. A comparison of the data in columns 6 and 5 (Table 16) shows that, with respect to pressed clay samples, the degeneracy of the silicon line intensity in fused clay samples with KI additives is enhanced.

 Consider SAR of fused clay sample 5 from clay with KI additives using the regression equation (5.2) obtained using pressed clay samples without KI additives as a base regression. From table 16 it is seen that sample 5 in relation to other clay samples with the addition of KI is the reference. For sample 5 and basic regression (5.2), the characteristic and fractional curves are constructed using dependence (4).

61,4912% С_М 62% ≠ C_siБ1, ω_о 0,9, ε= 10^-3.In the table. 17 shows data for estimating the value of C _si in sample 5 according to dependence (4) with the following initial data: I _si 0.9987 units; a -0.35573, b 0.0220264, C _siB1

61.4912% С _M 62% ≠ C _siБ1 , ω _о 0.9, ε = 10 ^-3 .

The fractional part of such quantities as a ₁ , b ₁ , I _A1 and I _A2 should contain a sufficient number of significant digits (digits).

According to the source data and the data of Table 17 for sample 5, the values of the coefficients В ₁ , В ₂ and В _{3 are} calculated for dependences (4) according to formulas (4.1) (4.3) (see table 18).

The number of significant digits of the fractional part of the quantities b ₁ , a ₁ , I _A1 and I _{A2 is} selected so that the number of significant digits of the fractional part of the coefficients B ₁ , B ₂ and B _{3 is} not less than 5, 6. The specified condition for the selected base regression is satisfied when the number of significant digits of the fractional part of the quantities b ₁ , a ₁ , I _A1 and I _{A2 is} not less than 9, 10.

Estimating the values of C _si for other values of ε by the above method, we obtain the dependence of C _si on ε, the graph of which is shown in Fig. 11. From Fig. 11 it is seen that significant perturbations of the value of C _si occur at ε <10 ⁻⁴ , therefore, ε _о <10 ⁻⁴ .

 In FIG. 12 shows the beat curve for sample 5 and the baseline regression obtained using extruded clay samples.

Given the possible content of "interfering" elements in the fused clay samples shown in Table 16, the initial data is selected for the search for С _sih in sample 5 (see table 19 and 20).

 In FIG. Figures 13, 14, and 15 show the output curves for estimating the silicon content in sample 5 using the base regression (5.2) in the first, second, and third approximations, respectively. The evaluation results are summarized in table.21.

 In addition to the data in Table 16, the contents of some “interfering” elements in fused clay samples with IR additives are given (Table 22).

As can be seen from tables 16 and 22, the most interesting from the point of view of the resolution of the method is sample 15, which contains a significant amount of “interfering” elements and the addition of quartz, while the dependence I _A f (a, b, C _{A is} used as the base regression ₎ ) obtained using pressed samples, and parameter K is a distinguishing feature.

 The initial data for evaluating the silicon content in sample 15 are given in tables 23 and 24.

 The results of estimates of the silicon content in sample 15 by the above method (using basic and auxiliary regressions) are given in table 25.

 In the table. Figure 26 shows the data for evaluating the silicon content in fused clay samples with the addition of quartz and the method error.

Let us consider the determination of the content of C _{AX of} silicon in fused samples composed of clay and Lyubertsy sand (PL) using a base regression defined as I _A a + bC _A.

 Table 27 shows data on the ratio of components (clay and PL) included in the samples, on the intensities of the silicon line, and the contents of the latter and “interfering” elements.

 As a reference sample for a number of samples shown in table 27, select sample 74 containing the smallest amount of PL.

The characteristic curve for sample 74 and basic regression I _A −0.35573 + +0.0220264 C _A obtained for silicon using pressed clay samples is shown in FIG. 15.

The fractional curve for sample 74 and basic regression with parameters a -0.35573 and b 0.0220264 at ε _o 8 · 10 ^-5 practically does not differ from that curve shown in Fig. 11.

 As an example, PCA gives an estimate of the silicon content in melted sample 62 (Table 27), which is of most interest due to the significant content of sand and "interfering" elements, in particular iron (9.54%). The degeneration of the intensity of the silicon line in sample 62 with respect to the base regression obtained using pressed clay samples is caused by the presence of PL, “interfering” elements, mainly iron and potassium, as well as by the difference in mineralogical composition (the analyzed sample is fused, and the basic samples pressed).

 The results of evaluations of the silicon content in sample 62 are presented in table 28.

 For the rest of the samples presented in table.27, the data on the estimates of the silicon content in them and the errors of the method are given in table.29.

In the table. 30 shows the results of evaluating the aluminum content in a fused perlite sample with an intensity of Y _AX 0.66912 using basic regression I _A = 0.023065 + + 0.051256 C _A , obtained on the basis of fused clay samples (ω _o 0.9; ε _o 8 · 10 ^-5 ; With _AB 13.504467%).

The aluminum content in perlite according to chemical analysis 12.56% therefore, the coefficient of variation K -0.09%
Consider a sample of the proposed SAR method of inhomogeneous materials in the case when there is no information on the direction of degeneracy of the intensity I _{AX of the} element being determined, for example, a perlite sample with a silicon content of _AX 73.24% and a silicon intensity I _AX 1.20609 units.

As can be seen from Fig. 17, equation (4) has two solutions, and in the absence of information on the direction of degeneracy of the intensity I _{AX, the} estimate of the content C _{AX of the} element being determined in the analyzed sample does not have a unique solution. In this case, the uncertainty in estimating the content of C _{AX of the} element _being determined is eliminated by using two basic regressions of type (2) obtained on the basis of two series of samples, each of which is characterized by a level of intensity degeneration I _A different from the other series of samples. The considered case of PCA occurs in rare cases of production problems, but is more common in research work.

= 73,16% при С_АХ 73,24%
Аналогичным способом оценивается содержание С_АХ определяемого элемента в образце, если базовые регрессии заданы двумя зависимостями типа (1,1) и отсутствует информация о направлении вырождения интенсивности I_АХ.In FIG. Figure 18 shows the results of evaluating the content of C _{AX of} silicon in a perlite sample using two basic regressions (2). As can be seen from Fig. 18, the solution to the problem has four values C _AX , two of which are almost identical in numerical value (points A and B '). The average value of the latter solutions is an estimate of the content of C _{AX of} silicon in the perlite sample and is equal to

= 73.16% at C _AX 73.24%
In a similar way, the content of С _{АХ of the} determined element in the sample is estimated if the base regressions are given by two dependencies of type (1,1) and there is no information about the direction of degeneration of intensity I _АХ .

 The above examples of the developed SAR method of non-homogeneous materials show that the method provides high accuracy (precision) in estimating the content of the element being determined in the analyzed sample, while the errors of the method are close to the quantization errors of samples of the analyzed materials; in the mechanization and automation of sample preparation processes, as well as in the computerization of mathematical calculations and the output of output data, the method is express, the total time for estimating the content of a determined element can be reduced according to preliminary data to 12-15 minutes (for pressed samples).

 For comparison, the data on the accuracy and expressivity of PCA according to the prototype: 3% relative errors and the total analysis time of 60 minutes

 The above are the results of the proposed PCA method for high element contents (from 12 to 80%). In the indicated range of element contents, PCA accuracy (relative errors) in the range of 1-3% was achieved. Errors of the known PCA methods for low element contents, in particular, lower than 2-3%, are especially large and make up 3-7% according to the expressness of the analysis. with the above achievements in terms of analysis accuracy, the total analysis time, depending on the method of sample preparation, ranges from 30 to 60 minutes or more. The reduction of the indicated time can be achieved (without fundamentally new solutions) only to the detriment of the accuracy of the analysis.

 Below are data on the assessment of low element concentrations, as well as proposals for the technology of practical use of the proposed method, based on data on the intensities and contents of Fe, K and Ca in clays and regression equations of type (2), obtained both by pressed and fused clay samples. The essence of the technology lies in the simultaneous use of these two regression equations, as well as auxiliary regression, compiled for one of these equations.

= 0,04+0,2 C_A
(6.1) по прессованным образцам и
C_A= -0,14+5,8•Y_A или

= 0,02702703+0,1930502 C_A
(6.2) по плавленным образцам глин.The following two independent regression equations of type (2) for Fe are obtained, for example, for a working range of Fe contents of 0.5-10%
C _A = -0.2 + 5 • Y _A or

= 0.04 + 0.2 C _A
(6.1) for pressed samples and
C _A = -0.14 + 5.8 • Y _A or

= 0.02702703 + 0.1930502 C _A
(6.2) for fused clay samples.

 The graphs (6.1) and (6.2) are similar to the graphs for silicon shown in Fig. 18.

-

)Шаг поиска равен ΔС_Мi/N, где N число подинтервалов, колеблющееся на практике от 10 до 16.In the presence of intensities Y _{AX of the} determined element A from fused samples, the regression equation obtained from pressed samples is used as the base regression. In this case, the direction of degeneracy of the intensity Y _AX with respect to the base (pressed) samples becomes known. The value C _{AX of the} analyzed element should be sought to the right of graph 1 (see Fig. 18). An auxiliary regression equation is compiled for an equation, for example, (6.1), and the content of the element being determined is searched for using the above scheme. To evaluate the interval and step h of the search for C _AX, C _A1 and C _A2 are evaluated according to equations (6.1) and (6.2). For the search interval C _AX, to a first approximation, we take a section adjacent to point C _AB1 C _A1 and equal to ΔC _Mi ≈ 2 (

-

) The search step is ΔС _Mi / N, where N is the number of sub- _ranges , which in practice _varies from 10 to 16.

If the intensity I _AX (Y _AX ) of the determined element A is obtained from pressed samples, then the equation obtained from fused samples is used as the basic regression equation. For the last equation, an auxiliary equation is compiled and the content of the element is searched.

The use of two equations of the type (6.1) and (6.2) makes it possible to increase the expressness of the analysis. This is achieved by the fact that, when searching for С _АХ1 in the first approximation, samples with the content of the element А С _АБ1 С _А2 , estimated, for example, according to equation (6.2), are used as comparison samples. In this case, two approximations are sufficient for estimating C _AX (taking into account the data of the shared curve).

 In the table. Figure 31 shows the results of evaluations of the contents of Fe, K, and Ca in clays by the proposed PCA method using the basic regression equations obtained from pressed samples.

 The above estimates of the contents of Fe, K, and Ca in clays were obtained practically with reliability at the level of quantization errors of samples (short-term apparatus errors KRF-IB). Given the achieved level of hardware errors of modern quantometers (0.05% or less), the developed SAR method is a technology that provides ultra-precise estimates of the content of elements in materials.

Claims (1)

METHOD FOR PRECISION EXPRESS X-RAY SPECTRAL ANALYSIS OF NON-HOMOGENEOUS MATERIALS, which consists in compiling from the initial samples made using the analyzed material, the initial dependence of the intensity I _{A of the} analytical line of the determined element A on its content C _A , in determining the intensities I _{A x} element A and in the analyzed _{a j} in the sample, a mixture of analysis sample and a predetermined material content C _{a B} and the element a quantitative comparison of the intensities I _{a x} and I _{a j} and using Khodnev depending I _A by C _A, characterized in that the produced basic samples which represent the original samples treated physically or made basic patterns of a material different mineralogical composition, wherein the intensity I _A baseline sample is degenerate with respect to the original samples, constitute basic dependence I _A on C _A , calculate the auxiliary dependence of I _A on C _A , which crosses the base, using a reference sample - a sample from the analyzed material, in which the content of element A is located in the lower boundary of the operating range of element A, according to the intensity I _{A 2} different from the base and initial samples, the optimal values of the parameters ε _o and ω _{o of the} auxiliary dependence of I _A on C _{A are} determined, where ε _o is the tangent of the angle between the auxiliary and I _a base dependencies of C _a, ω _o - optimum proportion of the analyzed samples in the estimated-sample mixtures from sample to be analyzed with a calculation of the reference sample and the sample to be analyzed with auxiliary comparison samples, the auxiliary sample - a sample, in a cat rum content of the element A is equal to the content of the element in the basic sample, while the intensity of line element A differs from the intensity of the latter is determined by calculation of the intensity I _{A Jan. 1} and I _{A February 1 i} a calculated sample-mixtures of the analyzed samples with the calculated base and subsidiary comparison samples, respectively, quantitatively compare the experimentally measured intensity I _{A x} with the calculated intensities I _{A 1 1} and I _{A 2 1 i} using at least two initial and basic dependences of I _A on C _A , according to the results of which the content is evaluated

element A in the analyzed sample in the first approximation, with the content of C _{A B 2} element A in the design samples of comparison, equal to

calculate the intensities I _{A 1 2} and I _{A 2 2 i} in the indicated sample mixtures and, based on the results of their quantitative comparison with the intensity I _{A x by the} above method, evaluate the content

element A in the second approximation, and in the estimation

in a third approximation, the content of C _{A B 3 of the} element A in the calculated comparison samples is equated with the estimate

obtained in the second approximation, and so on, until a constant value of C _{A x j is} obtained.

Images