Fractionation of chemical elements in the earth's crust

Fractionation of natural substances - the separation of elements from a single array under the influence of changes in the physico-chemical parameters of the surrounding medium. When analyzing fractionation, the behavior of at least two elements is considered.

General fractionation issues

The types of fractionation are distinguished:

Separation of elements occurs in the process of movement of a stream. As an example: separation of elements in a hydrothermal stream. Here, the elements are affected by both a change in flow parameters ( temperature ( T ), pressure ( P ), chemical potential), and kinetic parameters (for example, motion speed ) of the flow. As a result, in places there is a separation of the places of accumulation of elements, forming the so-called geochemical zoning. In establishing this zoning, an important role was played by the scientists of IMGRE: E. N. Baranov, A. I. Golovin, L. N. Ovchinnikov , S. V. Grigoryan et al. ^[1] , ^[2] .
Separation of elements under the influence of mineral formation conditions. The consequence of this is the redistribution of elements between minerals, and the nature of this process is fully described by the laws of thermodynamics . This allows the use of element concentrations in two coexisting minerals to assess the conditions of their formation .

The fractionation of stable isotopes of light elements has been studied to the greatest extent. A significant contribution to the solution of this problem was made by the Americans Bigeleysen ^[3] and Boting ^[4] . As applied to radiogenic elements (primarily to uranium and lead), some theoretical studies were carried out by HCUrey ^[5] , which revealed a weak effect on the separation of external environmental parameters and thereby vetoed their further study.

There is another difference between these systems: in stable isotope systems, all elements are rock-forming, reflecting the extreme case of isomorphism . This determines the possibility of their use for solving physico-chemical problems. In radiogenic systems, the daughter element is not an isotopic element relative to the parent isotope. All daughter elements, occupying various places in the periodic system of D. I. Mendeleev , differ from the mother in all respects, and primarily in size. Therefore, in addition to the influence of T, their distribution substantially depends on pressure and other physicochemical conditions of the mineral formation environment.

The problem of fractionation of radiogenic elements has been studied very poorly. G. For and D. Powell ^[6] noted the uniform distribution of radiogenic isotopes and isobars (RGII) in molten magma , which persists during crystallization , and discordance is associated with epigenetic processes . However, this statement, taken as an axiom , is not consistent with the phenomenon of fractionation of isomorphic and isotopic elements, which exhibit a physicochemical analogy with the Russian State Research Institute, taken into account during geobarothermometric studies.

Fractionation Levels

Two levels of studying fractionation can be distinguished.

The first level is due to a theoretical analysis of the conditions of this fractionation described in ^[7] . In Soviet geochemistry, these studies are presented primarily in the works of S. Z. Roginsky (1900-1970) ^[8] [1] and A. I. Brodsky http://www.warheroes.ru/hero/hero.asp? Hero_id = 12882 ( 06/19/1895 - 08/21/1969) ^[9] . Bigeleisen and Boting ^[10] brought these studies to their logical conclusion, that is, to the methodology for using them in practice . Concerning U and Pb theoretical studies were carried out only by HC Urey ^[11] .

The second level is due to the analysis of the actual distribution of the RGII in natural conditions.

The research results allowed us to identify groups of topics that reflect the possibility of fractionation of isotopes and isobars ^[12] .

Natural fractionation of radiogenic isotopes and isobars

It includes analysis of the distribution of radiogenic isotopes (isobars) in the exocontacts of intrusions, their joint behavior with stable isotopes of light elements and in individual minerals.

Fractionation in exocontacts of intrusions

Fig. 1: Dependence of the age of minerals determined by various methods on the distance to the intrusion contact in the contact metamorphism zone.

These studies were carried out mainly for stable isotopes of light elements (hereinafter referred to as SLE). The behavior of not only oxygen and carbon isotopes was studied, but also of Li , K (I. M. Morozova et al. ^[13] ), Mg and Ca (V. S. Lepin et al., 1969 ^[14] ; ^[15] ), B (Yu. P. Shergin et al. ^[16] ) and others. As a rule, Li and K are enriched with the light isotope of the central parts of the metasomatic zones and distillation of heavy differences to the edge parts. Mg and Ca have a clear dependence on the concentration of the element itself, in accordance with the Bachinsky rule ^[15] . Yu. P. Shergin and A. B. Kaminsky established a relative increase in the ¹¹ B isotope as they move away from the polymetallic ore body. A similar behavior is noted by T. E. Lovering ^[17] for the O isotope as it moves away from ore breccia. He also observed a decrease in the isotopic composition of C in calcite as it approached the intrusion.

As for radiogenic isotopes and isobars, such data is much less. E. L. Landa et al. ^[18] observed a change in Sr isotopes in apatites and apatite-bearing rocks of carbonatite complexes of the Kovdorsky and Gulinsky massifs. Hart S. R. ^[19] established pseudo-rejuvenation of age at the contact of the intrusions of Eldor and Oduban-Albia. The age of the Eldor intrusion according to BI according to the Ar - K method is estimated at 68 - 80 Ma. The age of the hornblende varies depending on the distance from the contact: at a distance of 1 - 76 m, it ranges from 120 to 1150 million years with a maximum of 1160 million years at around 41 m. A similar situation is noted for biotite near the Oduban-Albiya intrusion according to Rb- Sr - method; G. Sh. Ashkinadze also described similar situations ^[20] in the exocontact of the Ozernaya Varaka intrusion.

The behavior of Pb isotopes in the exocontact zones of the quartz-monzonite intrusion of Eldora Stock in Colorado was described by Dow BR et al. ^[21] . In orthoclases, not only the gross amount of Pb , but also the magnitude of the isotopic ratios changes: the ratio of ²⁰⁶ Pb / ²⁰⁴ Pb and ²⁰⁷ Pb / ²⁰⁴ Pb decreases significantly with distance from the contact. A detailed analysis of the behavior of isotopes in a thermal field was carried out by Hart SR ^[22] based on the study of biotite, feldspar (without specifying the species) and hornblendite using the Ar - K and Rb - Sr methods. According to these data, pseudo-rejuvenation of rocks occurs in almost all minerals in the contact zone, which must be considered as a manifestation of isobar migration in a temperature field.

Thus, the formation of the ratios of radiogenic isotopes and isobars is significantly affected by the temperature factor and very possibly pressure.

Co-distribution with stable isotopes of light elements (SLE)

In the analysis of SLEE, a significant influence on the distribution of temperature conditions of the formation of minerals was established. It is shown that in this case the distribution of isotopes of a pair of coexisting elements, for example, С - О (in calcite), Н - О (in mica), etc., or isotopes of one element in coexisting minerals, for example, for oxygen - Quartz - Biotite or sulfur in Galenite - Pyrite, in isothermal conditions it is described by the equation of a straight line ^[23] . When solving the inverse problem, if, under isothermal conditions, the distribution of isotopes paired with isotopes of a known element is described by a straight line equation as a reference, then we can talk about the effect of temperature on the distribution of isotopes of both elements. Therefore, in this case, the joint behavior of the RGII and SLEE in some temperature field is considered. In a relatively large number, the joint behavior of the ⁸⁷ Sr / ⁸⁶ Sr ratio and δ ¹⁸ O is described. Single works are known for the systems ⁱ Pb - S and (Ar-K) -δ ¹⁸ O.

The work carried out to study the joint behavior of strontium and oxygen isotopes in Costa Rica basalts (Barrett ^[24] ), in the kimberlites of Yakutia (Kostrovitsky ^[25] ), carbonatites (B. G. Pokrovsky et al. ^[26] ), smectites ^[27] minerals of Alp granites ^[28] , etc., as well as Pb and S isotopes in galena (Illinois, Kulp JL et al, ^[29] ; V.I. Vinogradov ^[30] , A.I. Tugarinov et al. ^{[31 ]} ) revealed a rather high correlation between the isotopes of these elements. Often, a direct relationship between ¹⁴ C and δ ¹³ C was described (V. Vinogradov ^[30] ; ^[32] ; and others).

In some studies, the isotopic composition of oxygen was compared with the age of rocks and minerals determined by the K-Ar method (Garlick et al. ^[33] ).

In all cases, the identification of linear relationships is explained solely by the phenomena of mixing (contamination) (for example, Kostrovitsky ^[25] ; A.A. Konev ^[34] ; Taylor ^[35] ). More plausible is the assumption that there is an isothermal redistribution of isotopes.

The effect of pressure is not clear. On isotopes whose dimensional parameters of atoms differ slightly, pressure weakly affects values up to 1 kbar. These conclusions are confirmed by experimental studies of RNClaton ^[36] and P. Harting ^[37] and others. Isobars are significantly different from each other, so the pressure on their distribution is significantly affected.

Mineral Fractionation

In geochronological equations, element contents are expressed by the number of atoms without specifying a unit of measurement , although more correctly, by the number of atoms in a unit volume of a substance. In modern analytics, the contents of elements are determined in relative units -%, g / t, etc. Therefore, the latter must be converted into a system of units of geochronological equations.
In the system of physical quantities, the main parameters characterizing the amount of a substance are mass (g) and volume (cm ³), and the quantity adequately reflecting these parameters is the density (or specific gravity ) d of this substance. Let N ^* be the number of atoms per unit volume, C is the relative concentration of this element in the compound, M is the mass of one atom of this substance, then N ^* = Cd / M. Since M does not fundamentally affect the subsequent conclusions, omitting it we get the equality N = Cd , showing the total mass of isotope atoms in unit volume. of spend for the isotope ²⁰⁶ Pb, for which we have $^{206}Pb=^{238}U\cdot {(e^{\lambda _{8}t}-1)}$ ${\ displaystyle ^ {206} Pb = ^ {238} U \ cdot {(e ^ {\ lambda _ {8} t} -1)}}$ . We will abbreviate this equation in the form

^{6}N=^{8}N\cdot {S_{o}}\qquad {(1)}

{\ displaystyle ^ {6} N = ^ {8} N \ cdot {S_ {o}} \ qquad {(1)}}

,

where ⁶ N is the number of ²⁰⁶ Pb isotope atoms formed during time t , ⁸ N is the number of ²³⁸ U uranium atoms remaining after decay; $\lambda _{8}$ ${\ displaystyle \ lambda _ {8}}$ - constant decay of uranium atoms ²³⁸ U ; S _{o is a} function of time. At t = const , equation (1) is an isochron equation with an angular coefficient S _o . In bilogarithmic coordinates, this equation takes the form:

ln{(^{6}N)}=ln{(^{8}N)}+ln{S_{o}}\qquad {(2)}

{\ displaystyle ln {(^ {6} N)} = ln {(^ {8} N)} + ln {S_ {o}} \ qquad {(2)}}

.

After transformations, equation (1) is reduced to the form

(^{6}C)\cdot {d}=(^{8}C)\cdot {d}\cdot {S_{o}}\qquad {(3)}

{\ displaystyle (^ {6} C) \ cdot {d} = (^ {8} C) \ cdot {d} \ cdot {S_ {o}} \ qquad {(3)}}

.

In the case of studying one sample, the value of d is reduced. However, for a reliable estimate of age ^[38], it is necessary to use two samples to construct isochrones with measured densities d ₁ and d ₂ . In this case, the angular coefficient S ^{* of the} quasi-isochron is determined from the equality

S^{*}={\frac {(^{6}C_{2}-^{6}C_{1})({\frac {d_{1}}{d_{2}}})}{(^{8}C_{2}-^{8}C_{1})({\frac {d_{1}}{d_{2}}})}}\qquad {(4)}

{\ displaystyle S ^ {*} = {\ frac {(^ {6} C_ {2} - ^ {6} C_ {1}) ({\ frac {d_ {1}} {d_ {2}}}) } {(^ {8} C_ {2} - ^ {8} C_ {1}) ({\ frac {d_ {1}} {d_ {2}}})}} qquad {(4)}}

This equality indicates the dependence of the isochron angular coefficient on the density of minerals. This position is illustrated in Table 1 and Fig. 2.

Fig. 2. Dependence of isotopic ratios on the density of the mineral. The numbers of the names of minerals are their density. Established by position quasi-isochron. Initial analyzes were taken from: 1- ^[39] ; 2- ^[40] ; 3— ^[41] ; 4 - ^[42] ; 5 - ^[43] . Minerals: AP-apatite; BI-biotite; DI-diopside; KSh-potassium feldspar, MN-monazite; MT magnetite; OL olivine; PL plagioclase; EN enstatitis.

Table 1. Dependence of isotopic ratios on the density of minerals in isobaric systems.
Minerals	Breeds	Raft- nost g / cm³	Content Relations isotopes			A source analyzes
Minerals	Breeds	Raft- nost g / cm³	Rb / sr	⁸⁷ Rb / ⁸⁶ Sr	⁸⁷ Sr / ⁸⁶ Sr	A source analyzes
Kalishpat	Syenite	2,56	0.1584	0.4587	0.70606	^[44]
Nepheline		2.60	0,0614	0.1777	0.70454
Arvfedsonite		3.45	0.0057	0.0166	0.70372
Sphen		3.56	0,0002	0,0007	0.70367
Kalishpat	Urtit	2,56	26.55	79.56	1,1121
Nepheline		2.60	2.61	7.69	0.744
Eudialyte		2.92	0.0012	0.0034	0.70386
Kalishpat	Metapelitis	2.59	0.102	0.468	0.71552	^[43]
Plagioclase	Metapelitis	2.76	0,030	0.0872	0.71532
Kalishpat	Granulitis	2,56	0.857	2.47	0.77341
Plagioclase	Granulitis	2.76	0.244	0.708	0.71980
Note: potassium feldspar - short for potassium feldspar.

Fig. 3: Distribution of the ratios of the concentrations of radiogenic isotopes and isobars between minerals. Isotopic systems : A-distribution of the ratio of the content of lead isotopes in minerals of the gneisses of Greenland ^[45] ; dependencies 1 - ²⁰⁶ Pb / ²⁰⁴ Pb; 2- ²⁰⁷ Pb / ²⁰⁴ Pb; 3- ²⁰⁸ Pb / ²⁰⁴ Pb. 4-observed straight lines, 5-supposed, based on the same age of Pb. B-ln ( ⁸⁷ Sr / ⁸⁶ Sr) in granites (C. Brooks ^[46] ). Isobaric systems : B-ln ( ⁸⁷ Sr / ⁸⁷ Rb) in granites ( ^[46] ). B-ln ( ⁴⁰ Ar / ⁴⁰ K) in granitoids and gneisses ^[47]

Additional information on the separation of isotopes and isobars is provided by an analysis of the distribution of isotopic (isobaric) relationships between minerals. An example of such distributions is shown in Fig. 2. In these cases, the experimental points are located on straight lines with an angular coefficient s ≠ 1 .

In practice, fractionation was indirectly illustrated by the series of distribution of ages by minerals and methods for determining age. For example, the sequences are constructed: for Karelia - PL (Rb-Sr)> MU (Rb-Sr)> MU (K-Ar) ≈ Mi (Rb-Sr)> BI (Rb-Sr) , where MI is microcline, MU- muscovite; for Finland, MI (Rb-Sr)> MD (Rb-Sr)> BI (Rb-Sr) ≈ BI (K-Ar) . More rigorously, this comparison is based on mineral comparisons of the ratios of the corresponding isotopes. As an example, table No. 2 shows some series in terms of the values of these relations:

Table 2. Private series of fractionation by the magnitude of relations.
System	Isotopic isobaric relations	Sequences minerals
Isotopic	²⁰⁶ Pb \ ²⁰⁴ Pb	SF> AP, MT> BI, PL> KSh
	²⁰⁷ Pb / ²⁰⁴ Pb	SF> AP, MT> BI, PL> KSh
	²⁰⁸ Pb / ²⁰⁴ Pb	SF> AP, MT> BI, PL> KSh
	⁸⁷ Sr / ⁸⁶ Sr	BI> KSh> PL
Isobaric	⁸⁷ Sr / ⁸⁷ Rb	PL> KSh> BI ≈ MU
Isobaric	⁴⁰ Ar / ⁴⁰ K	AM> BI> KSh> MU, BI> PL
Note: AM-Amphibole; SF-Sphene

The pattern of mineral distributions by these ratios is also revealed by comparing the ranked (in terms of density) d (reference) sequences of minerals arranged in decreasing density and those by isotopic (isobaric) ratios. In each pair of minerals, a mineral with a higher d value was put first. If in this case the isotopic (isobaric) ratios turned out to be similar to the ratios of the mineral densities, such pairs were called normal , otherwise inverse . Further, according to the ratio of normal and inverse pairs, general sequences of the arrangement of minerals were constructed. Comparison of these sequences with the reference was performed using the indicator (index) of the difference J ^[48] . The results of these comparisons are displayed in table No. 3 in the form of general sequences. For comparison, the sequence of minerals in terms of δ ¹⁸ O is shown.

Studies have shown that in isotope systems a heavy isotope accumulates in minerals with increased density, while in isobaric systems, isobar with a minimum size shows this tendency. In a more general case, an element with a higher atomic (ionic) density accumulates in a heavier mineral.

Table 3. General sequences of minerals in terms of ratios.
System	Isotope isobar relations	General sequence of minerals	J
Isotopic	Reference	УР> ГН> ПИ> МН> МТ> ЦР> ПХ> ОР> СФ> АП> БИ> КВ> ПЛ> КШ
	²⁰⁶ Pb \ ²⁰⁴ Pb	UR> (GN, PX)> MN ≈ CR> (OR, SF)> AP> MT> (PI, BI)> (HF, PL)> CS	0.13
	²⁰⁷ Pb / ²⁰⁴ Pb	УР> ГН> МН ≈ ЦР ≈ ОР> (ПХ, СФ)> МТ> АП> (ПИ, БИ)> (КВ, ПЛ)> КШ	0.15
	²⁰⁸ Pb / ²⁰⁴ Pb	(MN, OR)> [(SD ≈ GN), PR]> MT ≈ (ПХ, СФ)> АП ≈ (PI, BI)> (HF, PL)> КШ	0.13
	δ ¹⁸ O	HF> KS> PL> AM> BI> KP> OL> MT (acid rocks, ^[49] )	0.95
	δ ¹⁸ O	HF> KSh> MU> KI> AM> GR> BI> ChL> IL> MT (shales, ^[50] )	0.61
Isotopic	Reference	GR> SF> OL> KP> OP> AM> AP> BI> MU> PL> PL> NOT> KSh> SL	0
Isotopic	⁸⁷ Sr / ⁸⁶ Sr	(BI, OP)> MU> GR> (KSh, OL)> (KP, NOT, AM)> PL> AP> SF	0.37
Isobaric	⁸⁷ Sr / ⁸⁷ Rb	PL> AP> SF> (ME, AM)> KSh> MU> BI	0.33
	⁸⁷ Sr / ⁸⁷ Rb	KP> OP> OL> FM> BI	0.13
	⁴⁰ Ar / ⁴⁰ K	AM> MU> [NOT, (KP ≈ OP)]> (SD, HF)> BI> PL> KSh> FL	0.30
Note: AF-arvfedsonite; GL-galenite; GR-garnet; IL-ilmenite; KL-calcite; KB-quartz; KI-kyanite; KP-clinopyroxene; NOT-nepheline; OR-orthite; OP-orthopyroxene; PI-pyrite; PCP-pyrochlore; CD-sodalite; UR-uraninite; FL-phlogopite; CL-chlorite; CR-zircon; EV-eudialyte; EP-epidote .

Experimental fractionations

The whole complex of geological observations on the behavior of the Russian State Research Institute in a thermogradient field indicates the possibility of fractioning them under natural conditions. The overwhelming majority of studies came to this conclusion without mentioning the concept of “fractionation”. However, only experimental studies can make a final conclusion about the possibility of a phenomenon. Currently, the entire range of studies in this direction can be divided into two groups, which differ in the methodological methods of fractionation analysis:

Thermal heating of samples with analysis of the isotopic composition of the released product or sublimates;
Leaching (mainly lead isotopes) from natural formations under the influence of various reagents, often not directly related to the real conditions of isotope migration.

Analyzes were processed using an expression for the fractionation coefficient

\alpha ^{*}={\frac {(^{*}X/X)_{i}}{(^{*}X/X)_{o}}}

{\ displaystyle \ alpha ^ {*} = {\ frac {(^ {*} X / X) _ {i}} {(^ {*} X / X) _ {o}}}}

where ( ^* X / X) _o and ( ^* X / X) _i are the isotope ratios of the element X, the initial one after the experiment. The heavy isotope is marked with an index ( ^* ) . If isotopes of two elements X and Y are considered , then this expression is transformed into a working equation of the form

(^{*}X/X)_{m}=S^{*}\cdot {(^{*}Y/Y)_{n}}+F^{*}

{\ displaystyle (^ {*} X / X) _ {m} = S ^ {*} \ cdot {(^ {*} Y / Y) _ {n}} + F ^ {*}}

where m and n are some compounds. Often m = n . In this equation, the parameter S ^* = f (T) .

The purpose of these experiments: to identify the degree of preservation of isotope ratios in various thermodynamic conditions. Experiments are characterized by:

1. The experimental results are not considered from the position of fractionation of the RGII, which leads to ignoring the phenomenon of equilibrium distributions.
2. According to the theory of fractionation, it is necessary to study the isotopic composition of two compounds, but in these studies only one compound is considered. For example, when Pb is isolated from a mineral, the isotopic ratio is determined only in the sublimation and the composition of the residue from the sublimation is not analyzed. The same thing is noted with leaching: only the leach is analyzed, without touching the residue from leaching.
3.All experiments end with a qualitative statement of the results of changes in the isotope ratios without calculating the corresponding parameters: fractionation index, kinetic coefficients, etc.

Exposure to high temperatures

Lead isotope systems

I-Bar Systems K-Ar-

Isobaric Rb-Sr Systems

Leaching Impact

Fig. 4. The results of acid leaching of Pb isotopes: A-zircon (CR) [Paul et al. ^[51] ; Silver et. al ^[52] ], monazite (MN) (Sobotovich E.V. ^[53] ) granites. B - microcline and plagioclase of granites (Lobikov et al. ^[54] ). β _i = ( ⁱ Pb / ²⁰⁴ Pb) _mn / ( ⁱ Pb / ²⁰⁴ Pb) _pp , mn-mineral, pp-solution. B-fractionation of Pb isotopes between granite and accessory galena of Transbaikalia (Golubchina et al., ^[55] ). α _i = ( ⁱ Pb / ²⁰⁴ Pb) _g / ( ⁱ Pb / ²⁰⁴ Pb) _gn ; g-granite, gn-galenite.

Pb isotopes (about 92% of the studied samples), less often Sr-Rb isobars, and minimally K-Ar isobars were experimentally exposed. Pb isotopes were studied, as a rule, in accessory zircons and monazites, feldspars (usually potassium feldspars, plagioclases), biotites, uranite, granites, and other rocks and minerals. Sr-Rb isobars are in chondrite (Mittlefehldt DW et al ^[56] ), in basalt (Elderfild H, et al ^[57] ), K-Ar isobars are in biotite (Aprub S.V. ^[58] ), etc. d.

The main leaching agents are: nitric acid, less commonly HCl , HF and acetic , rarely distilled water. Acids - high concentrations up to concentrated, temperatures - more than 80 ° C. The leaching time ranged from the first hours to a month. Single samples were usually studied sporadically without observing the requirements for establishing isotopic equilibria.

The main goal of the research is to identify the degree of stability of the Russian State Research Institute in highly aggressive environments to establish the accuracy of determining the age of rocks. Systematic and targeted studies to identify the main patterns of migration of the Russian State Scientific Research Institute and their fractionation were not carried out. A generalization of these data was carried out ^[59] . Fragments of these studies are shown in Fig. 4. In generalizing, we used the idea of separation coefficients α in the form

\alpha _{i}={\left({\frac {^{i}Pb}{^{204}Pb}}\right)_{min}}/{\left({\frac {^{i}Pb}{^{204}Pb}}\right)_{s}}

{\ displaystyle \ alpha _ {i} = {\ left ({\ frac {^ {i} Pb} {^ {204} Pb}} \ right) _ {min}} / {\ left ({\ frac {^ {i} Pb} {^ {204} Pb}} \ right) _ {s}}}

where min is the investigated mineral, s is leached (the resulting solution) or another mineral; i = 206, 207, 208.

The data in Fig. 4 for accessory zircons and monazites (Fig. 4A) and feldspars (Fig. 4B) show the presence of certain regularities in the processes of redistribution of Pb isotopes between the studied mineral and the phase interacting with it, which are expressed in the linear behavior of the parameters lnα . Figure 4B shows a similar distribution of Pb isotopes between accessory galena and host granite. The presence of a similar linear relationship between the parameters lnα allows us to make an assumption about the existence of a geochemical isotope equilibrium between these substances.

Fractionation Modeling

When conducting experimental work of various types and levels, an addition or removal from the RGII system always occurs. This allows for a qualitative assessment of the impact of the input (removal) of the Russian State Institute of Geology to carry out numerical modeling. For this purpose, for a certain initial (reference) group of analyzes, for example, lead, with a known value of age t _et , a certain amount of lead isotopes is added, then, according to new data, age t ^{* is} calculated, which estimates the effect of adding isotope to the system with a reference. Then t _o is the age of the impurity lead; t _p is the age of the radiogenic additive. t ₁ , t ₂ and t ₃ - age, calculated respectively according to the equations:

^{206}Pb_{p}=^{238}U\cdot {f_{8}(t_{1})}

{\ displaystyle ^ {206} Pb_ {p} = ^ {238} U \ cdot {f_ {8} (t_ {1})}}

;

^{207}Pb_{p}=^{235}U\cdot {f_{5}(t_{2})}

{\ displaystyle ^ {207} Pb_ {p} = ^ {235} U \ cdot {f_ {5} (t_ {2})}}

;

{\gamma }_{p}=f(t_{3})={\frac {^{207}C_{p1}}{^{206}C_{p1}}}=R(U){\frac {f_{5}(t)}{f_{8}(t)}}

{\ displaystyle {\ gamma} _ {p} = f (t_ {3}) = {\ frac {^ {207} C_ {p1}} {^ {206} C_ {p1}}} = R (U) { \ frac {f_ {5} (t)} {f_ {8} (t)}}}

The mechanisms of changes in isochron parameters are distinguished:

Impurity factor - the value depends on the concentrations of Pb and isotope ratios (not studied);
An impurity factor is a constant value; it does not depend on Pb concentrations and isotopic ratios. This mechanism has been studied experimentally (numerical experiments) and theoretically.

The following factors were evaluated in the experiment:
one). Change in gross lead concentrations :

1a) Pb * = nPb (in the experiment, n = 0.5; 2). The effect on the parameters of equations (4) and (5) was found, but the age t of the lead Pb _o and Pb _p does not change.
1b) Pb * = Pb ± l ( l = 1; 2) affects the age t _o while maintaining t ₁ . With increasing l, the value of t _o increases in the case of (Pb + l) and decreases with (Pb - l) .

2). Change in the magnitude of the isotopic ratios X (= ²⁰⁶ Pb / ²⁰⁴ Pb ) and Y (= ²⁰⁷ Pb / ²⁰⁴ Pb ):

2a) relations of the type X * = Xβ _x ( β = 0.667; 0.833; 0.909; 1.1) are equivalent to the equality ⁱ C * = _i Ck _i ( Σk _i ≈ 4 and k _i = β _i (L / L *) , L and L * are the sums of the original and modified relationships, respectively). Changing X and Y changes all ages while maintaining the relationship between

t ₁ , t ₂ and t ₃ .

2b). X * = X ± l _x ( l = 10,20,50,100). Also ⁱ C * = ⁱ Ck _i in particular, β _x = (X + l _x ) / X. When changing X and Y changes t _o while maintaining t ₁ , t ₂ and t ₃ . The values of t _o increase with increasing l _y and decrease with increasing l _x .

Notes

↑ Barsukov V.L., Grigoryan S.V., Ovchinnikov L.N. Geochemical methods of prospecting ore deposits. M., Science, 1981.
↑ Baranov E.N. Endogenous geochemical halos of pyrite deposits. M., Science, 1987.
↑ Bigeleisen J. The effect of isotopic substitution on the entropy, enthalpy and heat capasity of ideal gases.// J. Chem. Phys. . 1953, 21, 8. P.1333-1339.
↑ Botinga J. Calculation of fractionation factors for carbon and oxygen isotopic exchange in the system calcite-carbon dioxide-water.// J. Phys. Chem. . 1968.72.3. P.800-808
↑ Urey HC // J.Chem.Soc. 1947.P.562
↑ Fore G., Powell D. Strontium Isotopes in Geology. M .: Mir, 1974.214 s.
↑ Bigeleisen J. The effect of isotopic substitution on the entropy, enthalpy and heat capasity of ideal gases.//J. Chem. Phys. 1953, 21, 8. P.1333-1339.
↑ Roginsky, S.Z., Theoretical Foundations of Isotopic Methods for the Study of Chemical Reactions, Moscow: Publishing House of the USSR Academy of Sciences. 1956 611 p.
↑ Brodsky A.I. Isotope Chemistry. M.: Publishing House of the Academy of Sciences of the USSR, 1957.
↑ Botinga J. Calculation of fractionation factors for carbon and oxygen isotopic exchange in the system calcite-carbon dioxide-water.//J. Phys. Chem .. 1968.72.3. P.800-808
↑ Urey HC "Research on the Natural Abundance of Deuterium and Other Isotopes in Nature. Final Report for Period Ending September 30, 1958
↑ Makarov V.P. Fractionation of radiogenic isotopes and isobars in natural conditions. // Domestic. Geology, 1993.8.P.63-71
↑ Morozova I.M. Alferovsky A.L., Yakovleva S.Z. Diffusion of Li and K isotopes in natural aluminosilicates. / Geochemistry of radiogenic and radioactive isotopes. L .: Nauka, 1974.P.105-130.
↑ Lepin V.S. Plyusnin G.S., Brandt S.B. Mass spectrometric analysis of Mg and Ca and their natural fractionation = isotors. / Yearbook, 1968. SB USSR Academy of Sciences. Irkutsk: 1969 ,. S. 2670 271.
↑ ¹ ² Plyusnin G.S., Brandt S. B. Isotopic fractionation of lithium, potassium, magnesium, calcium by zoning and paragenesis / Magmatism, formations of crystalline rocks and the depth of the Earth. Part 1.M .: Nauka, 1972. S.218-221
↑ Shergina Yu.P., Kaminskaya A.D. On the possibility of using natural variations of boron isotopes in geochemical searches. // Geochemistry, 1965, 1. P.64-67.
↑ Lovering T.S., McCarthy J.G., Friedman I. Value of ¹⁸ O / ¹⁶ O and ¹³ C / ¹² C ratios in hydrothermal dolomitic limestones and manganese carbonate metasomatic ores. / Chemistry of the Earth's crust, T.II. М.: Наука, 1964. С.616 - 629.
↑ Ланда Э.Л., Мурина Г.А., Шергина Ю.П., Краснова Н.И. Изотопный состав стронция в апататах и апатитоносных породах карбонатитовых комплексов.//Докл. АН СССР, 1982, 264, 6. С.1480-1482
↑ Харт С. Р. Возраст минералов и метаморфизм./Вопросы геохронологии. М.: изд-во ИЛ. 1980. С.45 −49
↑ Ашкинадзе Г.Ш. Миграция радиогенных изотопов в минералах. Л.: Наука, 1980. 144 с.
↑ Doe BR, Hart SR The effect of contact metamorphism on lead in potassium feld spars near the Eldora stock, Colorado.//J. Geophys.Res.,1963, 68, 11. P. 3511-3530.
↑ Hart SR The petrology and isotopic minerals age relation of a contact zone in the Front range, Colorado.//J. Geology, 1964, 72, 5. P. 493-525.
↑ Макаров В. П. Изотопные геотермометры./Мат-лы XIII научного семинара «Система планета Земля». М.: РОО «Гармония строения Земли и планет». 2005, С.93- 115.
↑ Barrett TJ, Friedrichsen H/ Strontium and oxygen isotopic composition of some basalts from Hole 504B, Costa Rica Rift, GSGP Legs 69 and 70.//Earth and Plenetary Sci.Let., 1982, V.60, 1. P. 27-38/
↑ ¹ ² Костровицкий С.И., Днепровская Л.В. и др. Корреляция изотопных составов Sr, C и O карбонатах из кимберлитов Якутии.//Докл. АН СССР, 1983, Т.272,5.С. 1223 - 1225.
↑ Покровский Б.Г.. Беляков А.Ю. и др. Происхождение карбонатитов и рудной толщи массива Томтор (СВ Якутия) по изотопным данным.//Геохимия, 1990, 9. С. 1320-1329.
↑ Standigel H. et al. Agents of low temperature Ocean Crust attaration.//Contr.Miner.Petrol., 1981, 77,3.P.150-157
↑ Haach U.,Hoefs J.,Gohn E. Constraits on the origin of Damarn granites by Rb/Sr and $\delta {^{18}O}$ $\delta {^{18}O}$ data.//Contr.Miner.Petrol.,1981, 79, 3/ P.279-289.
↑ Kulp JL et al. lead and sulfur isotopic abundance in Mississippi valley Galmas.//Bul.geol.Soc.Am.,1956, 67, 1. P.123- 124.
↑ ¹ ² Виноградов В.И. Распределение изотопов серы в минералах рудных месторождений./Изотопы серы и вопросы рудообразования. М.: Наука,1967.7-37.
↑ Тугаринов А.И., Митряева Н.М.. Занятин Н.И. и др. Изотопный состав свинца и серы и процесс рудообразования на месторождениях Атасуйского района.//Геохимия, 1982, 1972, 5. С. 547 - 561.
↑ Виноградов В.И. и др. ¹³ С/ ¹² С, ¹⁸ О/ ¹⁶ О и концентрация ¹⁴ С в карбонатитах вулкана Калианго (В.Африка).//Известия АН СССР, сер. геол., 1978,6. С.13-44.
↑ Garlick GD, Dymond JK Oxygen isotope exchange between volcanic materials and ocean water.//Bull. Geol.Soc.Amer., 1970,V.81, 7.P.2137- 2141
↑ Конев А.А., Воробьёв Е.И. и др. Об источниках вещества и генезис кальцитов в нефелинвых породах Прибайкалья по геохимическим и изотопным даннымю//Геохимия, 1984, 1. С. 50- 57.
↑ TaylorH.P. The effect of assimilation of country rock by magmas on $^{18}O/^{16}$ $^{18}O/^{16}$ and $^{87}Sr/^{Sr}$ $^{87}Sr/^{Sr}$ systematics in igneous rocks.//Earth and Plenetary.Sci.Let. 1980, V.47, 2. P.243 - 254
↑ Claton RN et al.Limits on the effect of pressure on isotopic fractionation.//Geoch.Cosmochym.Acta, 1975, 39, 8. P. 1197-1201.
↑ Harting P.Der thermodynamische kohlenstoffisotopiceffekt im system CH ₄ -H ₂ OPII//Isotopenprexis,1978, 14, 3/ P.99-101.
↑ Шуколюков Ю. А. и др. Графические методы изотопной геологии. М.: Наука, 1974.
↑ Zartman RE, Fer F. Lead concentration and isotopic composition in five peridotite inclusion of probable mantle origin.//Earth and Plenetary Sci.Let., 1973, 20, 1/ P. 54 - 66.
↑ Wanless RK, Stevens RD, Loveridge WD Anomalious parent- doughter isotope relationships in rocks adjacent to the Grenvill Front near Chibougamen, Quebec,//Eclogae Geol. Helv., 1970, 63, 1. P.345- 364.
↑ Гамильтон Е.И. Прикладная геохронология. М.: Недра,1968. 256 с.
↑ Aleinikoff JN, Zartman RE, Lyons JB U-Th-Pb geochronology of the massabasic gneiss and the granite, near Milford, South-Central New Hampshire: new evidence for Avalonian basement and Taconic and Alleghenian disturbances in Eastern New England.//Contrib.Miner.Petrol., 1979, 71, 1. P. 1 - 11
↑ ¹ ² Schenk VU-Pb and Rb-Sr radiometric dates and their correlation with metamorphic events in the granulite-facies basement of the Serre, Contheru Calabria (Italy).//Contrib.Miner.Petrol.,1980, 73, 1. P, 23 - 38.
↑ Когарко Л.Н., Крамм У., Грауэрт Б. Новые данные о возрасте и генезисе щелочных пород Ловозерского массива (изотопы рубидия и стронция).//Докл. АН СССР, 267, 4. С.970 - 972.
↑ Baadsgaard H., Lambert RSJ, Krupicka J. Mineral isotopic age relationships in the polymetamorphic Amitsog gneisses, Godthaab district, New Greenland.//Geochem.Cosmochem.Acta, 1978, 40, 5. P.513 - 527.
↑ ¹ ² Brooks C.The effect of mineral age discordancies on total rock Rb-Sr isochrones of the Heemskirt granite, Western Tasmania.//J.Geoph.Res., 1966,71,22.P.5447
↑ Aldrich LT, Davis GL, James HL Ages of minerals from metamorphic and igneous rocks near Iron Mountain, Michigan.//J. Petrology, 1965, 6, 3,P. 445- 472.
↑ Макаров В. П. Некоторые вопросы сравнения геохимических типов общих ореолов элементов рудных месторождений.//Геология и геофизика.1980.9.С.129-133
↑ Донцова Е.К. Изотопный обмен кислорода в процессе образования пород.//Геохимия, 1970, 8,. С.903 - 916.
↑ Garlick GD, Epstein S. oxygen isotope ratios in coexisting minerals of regionally metamorphosed rock.// Geochem.Cosmochem.Acta, 1967, 31, 2/ P.181- 214.
↑ Paul R., Howard AJ,Watson WW Isotopic termal-diffusion factor of argon.//J.Chem.Phys.,1963,39,11. P.53-56.1963
↑ Silver LT, Deutsch S. Uranium- lead isotopic variations in zircons: case study.//J/ Geol.,1963,71,6. P.721-758.
↑ Соботович Э. В. Свинцово-изохронное датирование горных пород./Вопросы прикладной геохимии. Киев: Наукова Думка, 1974. С.70-80
↑ Лобиков А.Ф., Овчинникова Л.В., Яковлева С.З. Изотопно-геохимические исследования гранитов Карташевского массива (Центр. Карелия). Новые данные о его генезисе и возрасте./Методические проблемы ядерной геологии. Л.: Наука, 1982.С.71.
↑ Голубчина М.Н.. Рабинович А.В. К вопросу о критериях связи оруденения с магматизмом по данным изогтопного анализа свинца.//Геохимия, 1957, 3. С.198-203.
↑ Mittlefehldt DW, Wetherill GW Rb-Sr studies of CJ and CM chondrites.//Geoch.Cosmochim. Acta, 1979,45,2. P.201-206.
↑ Elderfild H, Greaves MJ Strontium usotope gejchemistry of ocelanding geothermal system and implication for sea water chemystru.//Geoch.Cosmochim. Acta,1981, 45, 1. P.2201-2212.
↑ Апруб С.В. Влияние реакции изотопного обмена на K - Ar систему в минералах./Изотопный возраст горных пород и его геологическая интерпретация. Л.: Тр. ВСЕГЕИ, Т.328,1984. С.23-34.
↑ Макаров В. П. Основы теоретической геохронологии./Мат-лы XII научного семинара «Система планета Земля». М.: РОО «Гармония строения Земли и планет». 2004, С.228- 253.

[1] Barsukov V.L., Grigoryan S.V., Ovchinnikov L.N. Geochemical methods of prospecting ore deposits. M., Science, 1981.

[2] Baranov E.N. Endogenous geochemical halos of pyrite deposits. M., Science, 1987.

[Бигелейзен-3] Bigeleisen J. The effect of isotopic substitution on the entropy, enthalpy and heat capasity of ideal gases.// J. Chem. Phys. . 1953, 21, 8. P.1333-1339.

[Ботинга-4] Botinga J. Calculation of fractionation factors for carbon and oxygen isotopic exchange in the system calcite-carbon dioxide-water.// J. Phys. Chem. . 1968.72.3. P.800-808

[5] Urey HC // J.Chem.Soc. 1947.P.562

[6] Fore G., Powell D. Strontium Isotopes in Geology. M .: Mir, 1974.214 s.

[autogenerated2-7] Bigeleisen J. The effect of isotopic substitution on the entropy, enthalpy and heat capasity of ideal gases.//J. Chem. Phys. 1953, 21, 8. P.1333-1339.

[8] Roginsky, S.Z., Theoretical Foundations of Isotopic Methods for the Study of Chemical Reactions, Moscow: Publishing House of the USSR Academy of Sciences. 1956 611 p.

[9] Brodsky A.I. Isotope Chemistry. M.: Publishing House of the Academy of Sciences of the USSR, 1957.

[autogenerated1-10] Botinga J. Calculation of fractionation factors for carbon and oxygen isotopic exchange in the system calcite-carbon dioxide-water.//J. Phys. Chem .. 1968.72.3. P.800-808

[11] Urey HC "Research on the Natural Abundance of Deuterium and Other Isotopes in Nature. Final Report for Period Ending September 30, 1958

[Макаров-1-12] Makarov V.P. Fractionation of radiogenic isotopes and isobars in natural conditions. // Domestic. Geology, 1993.8.P.63-71

[13] Morozova I.M. Alferovsky A.L., Yakovleva S.Z. Diffusion of Li and K isotopes in natural aluminosilicates. / Geochemistry of radiogenic and radioactive isotopes. L .: Nauka, 1974.P.105-130.

[14] Lepin V.S. Plyusnin G.S., Brandt S.B. Mass spectrometric analysis of Mg and Ca and their natural fractionation = isotors. / Yearbook, 1968. SB USSR Academy of Sciences. Irkutsk: 1969 ,. S. 2670 271.

[Плюснин-15] ¹ ² Plyusnin G.S., Brandt S. B. Isotopic fractionation of lithium, potassium, magnesium, calcium by zoning and paragenesis / Magmatism, formations of crystalline rocks and the depth of the Earth. Part 1.M .: Nauka, 1972. S.218-221

[16] Shergina Yu.P., Kaminskaya A.D. On the possibility of using natural variations of boron isotopes in geochemical searches. // Geochemistry, 1965, 1. P.64-67.

[17] Lovering T.S., McCarthy J.G., Friedman I. Value of ¹⁸ O / ¹⁶ O and ¹³ C / ¹² C ratios in hydrothermal dolomitic limestones and manganese carbonate metasomatic ores. / Chemistry of the Earth's crust, T.II. М.: Наука, 1964. С.616 - 629.

[18] Ланда Э.Л., Мурина Г.А., Шергина Ю.П., Краснова Н.И. Изотопный состав стронция в апататах и апатитоносных породах карбонатитовых комплексов.//Докл. АН СССР, 1982, 264, 6. С.1480-1482

[19] Харт С. Р. Возраст минералов и метаморфизм./Вопросы геохронологии. М.: изд-во ИЛ. 1980. С.45 −49

[20] Ашкинадзе Г.Ш. Миграция радиогенных изотопов в минералах. Л.: Наука, 1980. 144 с.

[21] Doe BR, Hart SR The effect of contact metamorphism on lead in potassium feld spars near the Eldora stock, Colorado.//J. Geophys.Res.,1963, 68, 11. P. 3511-3530.

[22] Hart SR The petrology and isotopic minerals age relation of a contact zone in the Front range, Colorado.//J. Geology, 1964, 72, 5. P. 493-525.

[Макаров-3-23] Макаров В. П. Изотопные геотермометры./Мат-лы XIII научного семинара «Система планета Земля». М.: РОО «Гармония строения Земли и планет». 2005, С.93- 115.

[24] Barrett TJ, Friedrichsen H/ Strontium and oxygen isotopic composition of some basalts from Hole 504B, Costa Rica Rift, GSGP Legs 69 and 70.//Earth and Plenetary Sci.Let., 1982, V.60, 1. P. 27-38/

[Костровицкий-25] ¹ ² Костровицкий С.И., Днепровская Л.В. и др. Корреляция изотопных составов Sr, C и O карбонатах из кимберлитов Якутии.//Докл. АН СССР, 1983, Т.272,5.С. 1223 - 1225.

[26] Покровский Б.Г.. Беляков А.Ю. и др. Происхождение карбонатитов и рудной толщи массива Томтор (СВ Якутия) по изотопным данным.//Геохимия, 1990, 9. С. 1320-1329.

[27] Standigel H. et al. Agents of low temperature Ocean Crust attaration.//Contr.Miner.Petrol., 1981, 77,3.P.150-157

[28] Haach U.,Hoefs J.,Gohn E. Constraits on the origin of Damarn granites by Rb/Sr and $\delta {^{18}O}$ $\delta {^{18}O}$ data.//Contr.Miner.Petrol.,1981, 79, 3/ P.279-289.

[29] Kulp JL et al. lead and sulfur isotopic abundance in Mississippi valley Galmas.//Bul.geol.Soc.Am.,1956, 67, 1. P.123- 124.

[Виноградов-30] ¹ ² Виноградов В.И. Распределение изотопов серы в минералах рудных месторождений./Изотопы серы и вопросы рудообразования. М.: Наука,1967.7-37.

[31] Тугаринов А.И., Митряева Н.М.. Занятин Н.И. и др. Изотопный состав свинца и серы и процесс рудообразования на месторождениях Атасуйского района.//Геохимия, 1982, 1972, 5. С. 547 - 561.

[32] Виноградов В.И. и др. ¹³ С/ ¹² С, ¹⁸ О/ ¹⁶ О и концентрация ¹⁴ С в карбонатитах вулкана Калианго (В.Африка).//Известия АН СССР, сер. геол., 1978,6. С.13-44.

[33] Garlick GD, Dymond JK Oxygen isotope exchange between volcanic materials and ocean water.//Bull. Geol.Soc.Amer., 1970,V.81, 7.P.2137- 2141

[34] Конев А.А., Воробьёв Е.И. и др. Об источниках вещества и генезис кальцитов в нефелинвых породах Прибайкалья по геохимическим и изотопным даннымю//Геохимия, 1984, 1. С. 50- 57.

[35] TaylorH.P. The effect of assimilation of country rock by magmas on $^{18}O/^{16}$ $^{18}O/^{16}$ and $^{87}Sr/^{Sr}$ $^{87}Sr/^{Sr}$ systematics in igneous rocks.//Earth and Plenetary.Sci.Let. 1980, V.47, 2. P.243 - 254

[36] Claton RN et al.Limits on the effect of pressure on isotopic fractionation.//Geoch.Cosmochym.Acta, 1975, 39, 8. P. 1197-1201.

[37] Harting P.Der thermodynamische kohlenstoffisotopiceffekt im system CH ₄ -H ₂ OPII//Isotopenprexis,1978, 14, 3/ P.99-101.

[38] Шуколюков Ю. А. и др. Графические методы изотопной геологии. М.: Наука, 1974.

[39] Zartman RE, Fer F. Lead concentration and isotopic composition in five peridotite inclusion of probable mantle origin.//Earth and Plenetary Sci.Let., 1973, 20, 1/ P. 54 - 66.

[40] Wanless RK, Stevens RD, Loveridge WD Anomalious parent- doughter isotope relationships in rocks adjacent to the Grenvill Front near Chibougamen, Quebec,//Eclogae Geol. Helv., 1970, 63, 1. P.345- 364.

[41] Гамильтон Е.И. Прикладная геохронология. М.: Недра,1968. 256 с.

[42] Aleinikoff JN, Zartman RE, Lyons JB U-Th-Pb geochronology of the massabasic gneiss and the granite, near Milford, South-Central New Hampshire: new evidence for Avalonian basement and Taconic and Alleghenian disturbances in Eastern New England.//Contrib.Miner.Petrol., 1979, 71, 1. P. 1 - 11

[Schenk-43] ¹ ² Schenk VU-Pb and Rb-Sr radiometric dates and their correlation with metamorphic events in the granulite-facies basement of the Serre, Contheru Calabria (Italy).//Contrib.Miner.Petrol.,1980, 73, 1. P, 23 - 38.

[44] Когарко Л.Н., Крамм У., Грауэрт Б. Новые данные о возрасте и генезисе щелочных пород Ловозерского массива (изотопы рубидия и стронция).//Докл. АН СССР, 267, 4. С.970 - 972.

[45] Baadsgaard H., Lambert RSJ, Krupicka J. Mineral isotopic age relationships in the polymetamorphic Amitsog gneisses, Godthaab district, New Greenland.//Geochem.Cosmochem.Acta, 1978, 40, 5. P.513 - 527.

[Brooks-46] ¹ ² Brooks C.The effect of mineral age discordancies on total rock Rb-Sr isochrones of the Heemskirt granite, Western Tasmania.//J.Geoph.Res., 1966,71,22.P.5447

[47] Aldrich LT, Davis GL, James HL Ages of minerals from metamorphic and igneous rocks near Iron Mountain, Michigan.//J. Petrology, 1965, 6, 3,P. 445- 472.

[Макаров-2-48] Макаров В. П. Некоторые вопросы сравнения геохимических типов общих ореолов элементов рудных месторождений.//Геология и геофизика.1980.9.С.129-133

[49] Донцова Е.К. Изотопный обмен кислорода в процессе образования пород.//Геохимия, 1970, 8,. С.903 - 916.

[50] Garlick GD, Epstein S. oxygen isotope ratios in coexisting minerals of regionally metamorphosed rock.// Geochem.Cosmochem.Acta, 1967, 31, 2/ P.181- 214.

[51] Paul R., Howard AJ,Watson WW Isotopic termal-diffusion factor of argon.//J.Chem.Phys.,1963,39,11. P.53-56.1963

[52] Silver LT, Deutsch S. Uranium- lead isotopic variations in zircons: case study.//J/ Geol.,1963,71,6. P.721-758.

[53] Соботович Э. В. Свинцово-изохронное датирование горных пород./Вопросы прикладной геохимии. Киев: Наукова Думка, 1974. С.70-80

[54] Лобиков А.Ф., Овчинникова Л.В., Яковлева С.З. Изотопно-геохимические исследования гранитов Карташевского массива (Центр. Карелия). Новые данные о его генезисе и возрасте./Методические проблемы ядерной геологии. Л.: Наука, 1982.С.71.

[55] Голубчина М.Н.. Рабинович А.В. К вопросу о критериях связи оруденения с магматизмом по данным изогтопного анализа свинца.//Геохимия, 1957, 3. С.198-203.

[56] Mittlefehldt DW, Wetherill GW Rb-Sr studies of CJ and CM chondrites.//Geoch.Cosmochim. Acta, 1979,45,2. P.201-206.

[57] Elderfild H, Greaves MJ Strontium usotope gejchemistry of ocelanding geothermal system and implication for sea water chemystru.//Geoch.Cosmochim. Acta,1981, 45, 1. P.2201-2212.

[58] Апруб С.В. Влияние реакции изотопного обмена на K - Ar систему в минералах./Изотопный возраст горных пород и его геологическая интерпретация. Л.: Тр. ВСЕГЕИ, Т.328,1984. С.23-34.

[Макаров-4-59] Макаров В. П. Основы теоретической геохронологии./Мат-лы XII научного семинара «Система планета Земля». М.: РОО «Гармония строения Земли и планет». 2004, С.228- 253.