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Radiation enhanced diffusion: approximate solutions of chemical rate equations and their application to experiments
Journal of Nuclear Materials 82 (1979) 298-301 0 North-Holland Publishing Company
RADIATION ENHANCED DIFFUSION: APPROXIMATE SOLUTIONS OF CHEMICAL RATE EQUATIONS AND THEIR APPLICATION TO EXPERIMENTS C. ABROMEIT and R. POERSCHKE Hahn-Meitner-Institut fiir Kernforschung, Arbeitsgruppe C 2, Glienicker Str. 100, 1000 Berlin 39, Germany Received 19 January 1979
Analytical solutions for the time dependence of point defect concentrations in radiation enhanced diffusion experiments are presented, including good approximations for special cases of defect reactions. An instruction for the analysis of experimental data is given for determining reaction cases and physical parameters such as capture radii and interstitial migration energy.
1. Introduction
dences with slopes different from (0, ff , 1) which can be interpreted as a transition behaviour between different reaction regimes [4,6]. A detailed study, especially of the transient regimes, is necessary for the interpretation of the experimental results. For the above reasons, approximative analytic solutions of the chemical rate equations were derived for all transient regimes for the concentrations of single vacancies and interstitials. Moreover a simple procedure for the evaluation of the physical parameters from the experimental data will be given.
The time evolution of radiation enhanced diffusion in metals and alloys depends on physical parameters such as migration energies of vacancies (v), E,“, and interstitials (i), fl, capture radii for freely migrating interstitials at sinks, ris, or at vacancies, qv. Therefore, it should be principally possible to determine these parameters from experiments in which the time dependence of the atomic jump rate was measured, e.g., Zener relaxation [l] or resistivity measurements [2-41, Unfortunately, the time resolution of the experimental methods applied is not sufficient and/or the measurements have not been extended sufficiently in order to allow an adequate evaluation of the above parameters. This problem can be overcome when the theoretical time evolution is determined not only for time periods in which asymptotic laws are valid but also for transition periods between the asymptotic intervals. Sizmann has recently reviewed the theoretical treatments of radiation enhanced diffusion [S]. The results can be summarized as follows: The timedependence of the point defect concentrations c are divided in regions of different time behaviour limited by characteristic times. In a plot of log c vs. log t the solutions for c appear as straight lines with slopes of e.g. (0, +3, +l) in different reaction regimes. No analytic expressions exist for the transitions of one region to another. But some experimental data clearly show time depen-
2. Kinetic equations We consider two defect species, vacancies v and interstitials i which are both produced at a rate ke. They may lead to the following defect reactions: recombination
itv-+O,
annealing at sinks
its+O, v+s+o.
The chemical rate equations for these reactions d zcv d P
= ko - ki&iCv - kvscscv,
(1) . = ko - kivcicv - ki,csci
contain the rate constants kiv for recombination and 298
ki,, k, for annealing at sinks, where we have followed the notation of Sizm~n. For our calculations, the mean defect separation was assumed to be larger than the mean distance between extended sinks (surfaces) and therefore a diffusion term VDVc was omitted [5]. The rate constants are given by ki, = clWi,(Di +Dv)/SJ, ki, = 4rGsDiJS1,
(2)
k vs = 4nr,,D,&& for a diffusion controlled reaction (riv recombination radius; G,, r, trapping radii for i, v at sinks; Di, Dv diffusion constants for i, v; 51 voiume of a lattice atom). Because of the different migration energies for vacancies and interstitials E,” > EF, the rate constant kt, is much smalIer than kt, and ki,. Analytic approximations of the solution of eqs. (1) becomes important if mutual recombination happens and sinks contribute to interstitial annihilation too. In this case, there is a region of time dependence cv - t’j2 and Ci - f-rj2 for temperatures and times where the vacancy sink term in eqs. (1) can be neglected (&oki,/ki,kwCz > I). This is true for low temperatures and for small sink densities.
3. Solutions of the chemical rate equations The results of the calculations are shown in figs. 1 and 2 in a log-log plot. The curves correspond to two different cases in the build-up of the concentrations. Either recombination of ~terstiti~ with vacancies (fig. 1) or interstitial migration to sinks (fig. 2) dominate after the linear defect production regime. Quantitatively, these cases are characterized by the ratio
We will briefly give the mean features of the time evolution of the concentrations. For A > 1 (recombination case) mutual recombination dominates first (region II) with a time constant tl After this quasi-stationa~ regime at t2, sinks contribute additiona~y to the Interstitial annihilation (region III). For A < 1 (sink case), the influence of sinks is important earlier than recombination with different time constants tl and t2 but the same time depen-
log t
Ci =
(“‘Ki,5i2. ( 1+ OL(1).t/+,)-“’
with a(t) z e IU-IL? : Cu= (+$-r ti = ($+)“2. If
(1-exp (-t/t,))1’2
(1- exp (-t/t,)
)-“2
Fi. I. Time evolution of defect concentrations during irradiation for the recombination case A > I.
dence turns out for region III in both cases. The transition to region IV depends on the migration energy of vacancies and is equal in both cases A ;B 1. The analytic fo~ulae were found by the fo~ow~g procedure: The rate equations can be readily integrated in the regime I-II. For A < 1 and ki, = 0 (neglecting any recomb~ation), they become uncoupled and for A > 1 and kg, = k, = 0 (no influence of sinks) identicai equations follow for ci and c,. In the transition regime II-III for A < 1, the analytic expressions of Poerschke and Wo~enberger [4J were confIrmed. In alI other cases (A > 1; II-III and A zt 1; III-IV), equations were derived by a systematic approximation of numerical solution of the chemical rate equations. The GEAR package of subroutines, developed at Lawrence Iivermore’Laboratory 171, was used for the numerical integration. .The temperature T = 50-800 R, sink density c, = 10-s-lO”o and migration energy of ~te~titi~s Z$“ = 0.1-0.5 eV
C. Abromeit. R. Poerschke / R~d~ut~n enhanced d~f~sion
i,
teq
i,
log t A
=e
t, = I-II
:
1
“is%
< t* =
B
< t,, 0
L:
I”
_.I-_ 2%,c,
cv = K,t
Fig. 3. The different curvatures for A S 1 in a plot D&t versusy2 = D?,d. The values m, y11 and tz are chosen according to eq. (5) in the text.
c, = (Ko/KigC5).(1-exp(-t/t,l) 1-m :
t,
q
(K,,c,/KiV).[(lct/tz)!‘2-t1
Ci = (Ko/i<,,c,).fl+tlt2J“”
ti = (ggf
(1-exp(-tite,)
Fii. 2. Time evolution of defect concentrations tion for the sink case A < 1.
j”2
during icradia-
were varied over a wide range to assure the validity of the given approximations. In all cases the error was less than 4%. Generally, the asymptotic form (t + 0, t + -) of our solutions (straight lines) are in principal accordance with those given by Sizmann, but the solutions for cv and ci in region III differ from those in ref. [S] by a factor of 2.
4. Conclusions for the analysis of experiments on radiation enhanced diffusion The given formulae can be helpful for the analysis of measured radiation enhanced diffusion constants &j = C l)jcj. If the experiments give only the time dependence of &$ in one of the regions I-IV, a variation of parameters such as tem~ra~e T, point defect production rate kO, or sink den~ty c, are necessary to get information about the capture radii Q~, Q or migration energy Ef” . The analysis becomes diff-
cult especially in the region III because of the equal beha~our for A P 1. But in some experiments, e.g., electrical resistivity measurements, a beginning curvature to region II can be seen [4,6]. From this we are now able to decide whether we have the sinks or the recombination being dominating and to evaluate the solution in the neighbouring region. In the regions II and III, we can neglect the influence of the vacancies. The radiation enhanced diffusion constant &ad becomes
Dr&(t) =DiCi(t> =Y.
(4) The difference between the two curvatures for A Z?1 can be shown clearly by a plot D&d(t) t versus I)&d (f). Fig. 3 gives the two cases. The end points of the curves are the solutions in region III (&d(?) = 0) and in the region II (D$(t) t = 0). The solution for the sink case is a straight line; for the recombination case, there is a parabola with its maximum at D& (t> = $ Y~Iand t”
lots. The
values of m2 and yf~ are:
+g$, iv @
18
s
&JkiscsY t
forA < 1,
YZl= oi’ @&iv),
forA > 1.
(5)
C. Abromeit, R. Poerschke /Radiation enhanced diffusion
The important difference therefore comes from the beginning curvature for long time t > t2. A plot of the experimental data in the way described above immediately gives the reaction case A 2 1. The values of y$ can be obtained if the maximum value for A > 1 is identified. The described procedure gives the physical parameters in a very simple manner, even if only a small part of the time evolution of the radiation enhanced diffusion coefficient is measured. Acknowledgement The authors are grateful to Prof. Dr. H. Wollenberger for detailed discussions and for reading the
301
manuscript. They also wish to thank Mrs. S. Franke for her support during the numerical calculations.
References [l] M. HaIbwachs, H. Gonzalez and J. HiIlairet, Scripta Met. 9 (1975) 575. [2] L.N. Bystrov, L.I. Ivanov and Y.M. Platov, Phys. Stat. Sol. (a) 8 (1971) 375. [ 31 R. Poerschke and H. WoIlenberger, J. Nucl. Mater. 74 (1978) 48. [4] R. Poerschke and H. WoIIenberger, J. Phys. F6 (1976) 27. [S] R. Sizmann, J. Nucl. Mater. 69 & 70 (1978) 386. [ 61 R. Poerschke and H. WoIlenberger, to be published. [ 71 A.C. Hindmarsh, Lawrence Livermore Laboratory Report UCID-3000 (1974).
RADIATION ENHANCED DIFFUSION: APPROXIMATE SOLUTIONS OF CHEMICAL RATE EQUATIONS AND THEIR APPLICATION TO EXPERIMENTS C. ABROMEIT and R. POERSCHKE Hahn-Meitner-Institut fiir Kernforschung, Arbeitsgruppe C 2, Glienicker Str. 100, 1000 Berlin 39, Germany Received 19 January 1979
Analytical solutions for the time dependence of point defect concentrations in radiation enhanced diffusion experiments are presented, including good approximations for special cases of defect reactions. An instruction for the analysis of experimental data is given for determining reaction cases and physical parameters such as capture radii and interstitial migration energy.
1. Introduction
dences with slopes different from (0, ff , 1) which can be interpreted as a transition behaviour between different reaction regimes [4,6]. A detailed study, especially of the transient regimes, is necessary for the interpretation of the experimental results. For the above reasons, approximative analytic solutions of the chemical rate equations were derived for all transient regimes for the concentrations of single vacancies and interstitials. Moreover a simple procedure for the evaluation of the physical parameters from the experimental data will be given.
The time evolution of radiation enhanced diffusion in metals and alloys depends on physical parameters such as migration energies of vacancies (v), E,“, and interstitials (i), fl, capture radii for freely migrating interstitials at sinks, ris, or at vacancies, qv. Therefore, it should be principally possible to determine these parameters from experiments in which the time dependence of the atomic jump rate was measured, e.g., Zener relaxation [l] or resistivity measurements [2-41, Unfortunately, the time resolution of the experimental methods applied is not sufficient and/or the measurements have not been extended sufficiently in order to allow an adequate evaluation of the above parameters. This problem can be overcome when the theoretical time evolution is determined not only for time periods in which asymptotic laws are valid but also for transition periods between the asymptotic intervals. Sizmann has recently reviewed the theoretical treatments of radiation enhanced diffusion [S]. The results can be summarized as follows: The timedependence of the point defect concentrations c are divided in regions of different time behaviour limited by characteristic times. In a plot of log c vs. log t the solutions for c appear as straight lines with slopes of e.g. (0, +3, +l) in different reaction regimes. No analytic expressions exist for the transitions of one region to another. But some experimental data clearly show time depen-
2. Kinetic equations We consider two defect species, vacancies v and interstitials i which are both produced at a rate ke. They may lead to the following defect reactions: recombination
itv-+O,
annealing at sinks
its+O, v+s+o.
The chemical rate equations for these reactions d zcv d P
= ko - ki&iCv - kvscscv,
(1) . = ko - kivcicv - ki,csci
contain the rate constants kiv for recombination and 298
ki,, k, for annealing at sinks, where we have followed the notation of Sizm~n. For our calculations, the mean defect separation was assumed to be larger than the mean distance between extended sinks (surfaces) and therefore a diffusion term VDVc was omitted [5]. The rate constants are given by ki, = clWi,(Di +Dv)/SJ, ki, = 4rGsDiJS1,
(2)
k vs = 4nr,,D,&& for a diffusion controlled reaction (riv recombination radius; G,, r, trapping radii for i, v at sinks; Di, Dv diffusion constants for i, v; 51 voiume of a lattice atom). Because of the different migration energies for vacancies and interstitials E,” > EF, the rate constant kt, is much smalIer than kt, and ki,. Analytic approximations of the solution of eqs. (1) becomes important if mutual recombination happens and sinks contribute to interstitial annihilation too. In this case, there is a region of time dependence cv - t’j2 and Ci - f-rj2 for temperatures and times where the vacancy sink term in eqs. (1) can be neglected (&oki,/ki,kwCz > I). This is true for low temperatures and for small sink densities.
3. Solutions of the chemical rate equations The results of the calculations are shown in figs. 1 and 2 in a log-log plot. The curves correspond to two different cases in the build-up of the concentrations. Either recombination of ~terstiti~ with vacancies (fig. 1) or interstitial migration to sinks (fig. 2) dominate after the linear defect production regime. Quantitatively, these cases are characterized by the ratio
We will briefly give the mean features of the time evolution of the concentrations. For A > 1 (recombination case) mutual recombination dominates first (region II) with a time constant tl After this quasi-stationa~ regime at t2, sinks contribute additiona~y to the Interstitial annihilation (region III). For A < 1 (sink case), the influence of sinks is important earlier than recombination with different time constants tl and t2 but the same time depen-
log t
Ci =
(“‘Ki,5i2. ( 1+ OL(1).t/+,)-“’
with a(t) z e IU-IL? : Cu= (+$-r ti = ($+)“2. If
(1-exp (-t/t,))1’2
(1- exp (-t/t,)
)-“2
Fi. I. Time evolution of defect concentrations during irradiation for the recombination case A > I.
dence turns out for region III in both cases. The transition to region IV depends on the migration energy of vacancies and is equal in both cases A ;B 1. The analytic fo~ulae were found by the fo~ow~g procedure: The rate equations can be readily integrated in the regime I-II. For A < 1 and ki, = 0 (neglecting any recomb~ation), they become uncoupled and for A > 1 and kg, = k, = 0 (no influence of sinks) identicai equations follow for ci and c,. In the transition regime II-III for A < 1, the analytic expressions of Poerschke and Wo~enberger [4J were confIrmed. In alI other cases (A > 1; II-III and A zt 1; III-IV), equations were derived by a systematic approximation of numerical solution of the chemical rate equations. The GEAR package of subroutines, developed at Lawrence Iivermore’Laboratory 171, was used for the numerical integration. .The temperature T = 50-800 R, sink density c, = 10-s-lO”o and migration energy of ~te~titi~s Z$“ = 0.1-0.5 eV
C. Abromeit. R. Poerschke / R~d~ut~n enhanced d~f~sion
i,
teq
i,
log t A
=e
t, = I-II
:
1
“is%
< t* =
B
< t,, 0
L:
I”
_.I-_ 2%,c,
cv = K,t
Fig. 3. The different curvatures for A S 1 in a plot D&t versusy2 = D?,d. The values m, y11 and tz are chosen according to eq. (5) in the text.
c, = (Ko/KigC5).(1-exp(-t/t,l) 1-m :
t,
q
(K,,c,/KiV).[(lct/tz)!‘2-t1
Ci = (Ko/i<,,c,).fl+tlt2J“”
ti = (ggf
(1-exp(-tite,)
Fii. 2. Time evolution of defect concentrations tion for the sink case A < 1.
j”2
during icradia-
were varied over a wide range to assure the validity of the given approximations. In all cases the error was less than 4%. Generally, the asymptotic form (t + 0, t + -) of our solutions (straight lines) are in principal accordance with those given by Sizmann, but the solutions for cv and ci in region III differ from those in ref. [S] by a factor of 2.
4. Conclusions for the analysis of experiments on radiation enhanced diffusion The given formulae can be helpful for the analysis of measured radiation enhanced diffusion constants &j = C l)jcj. If the experiments give only the time dependence of &$ in one of the regions I-IV, a variation of parameters such as tem~ra~e T, point defect production rate kO, or sink den~ty c, are necessary to get information about the capture radii Q~, Q or migration energy Ef” . The analysis becomes diff-
cult especially in the region III because of the equal beha~our for A P 1. But in some experiments, e.g., electrical resistivity measurements, a beginning curvature to region II can be seen [4,6]. From this we are now able to decide whether we have the sinks or the recombination being dominating and to evaluate the solution in the neighbouring region. In the regions II and III, we can neglect the influence of the vacancies. The radiation enhanced diffusion constant &ad becomes
Dr&(t) =DiCi(t> =Y.
(4) The difference between the two curvatures for A Z?1 can be shown clearly by a plot D&d(t) t versus I)&d (f). Fig. 3 gives the two cases. The end points of the curves are the solutions in region III (&d(?) = 0) and in the region II (D$(t) t = 0). The solution for the sink case is a straight line; for the recombination case, there is a parabola with its maximum at D& (t> = $ Y~Iand t”
lots. The
values of m2 and yf~ are:
+g$, iv @
18
s
&JkiscsY t
forA < 1,
YZl= oi’ @&iv),
forA > 1.
(5)
C. Abromeit, R. Poerschke /Radiation enhanced diffusion
The important difference therefore comes from the beginning curvature for long time t > t2. A plot of the experimental data in the way described above immediately gives the reaction case A 2 1. The values of y$ can be obtained if the maximum value for A > 1 is identified. The described procedure gives the physical parameters in a very simple manner, even if only a small part of the time evolution of the radiation enhanced diffusion coefficient is measured. Acknowledgement The authors are grateful to Prof. Dr. H. Wollenberger for detailed discussions and for reading the
301
manuscript. They also wish to thank Mrs. S. Franke for her support during the numerical calculations.
References [l] M. HaIbwachs, H. Gonzalez and J. HiIlairet, Scripta Met. 9 (1975) 575. [2] L.N. Bystrov, L.I. Ivanov and Y.M. Platov, Phys. Stat. Sol. (a) 8 (1971) 375. [ 31 R. Poerschke and H. WoIlenberger, J. Nucl. Mater. 74 (1978) 48. [4] R. Poerschke and H. WoIIenberger, J. Phys. F6 (1976) 27. [S] R. Sizmann, J. Nucl. Mater. 69 & 70 (1978) 386. [ 61 R. Poerschke and H. WoIlenberger, to be published. [ 71 A.C. Hindmarsh, Lawrence Livermore Laboratory Report UCID-3000 (1974).