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Small-angle scattering of X-rays by electron-irradiated copper
Volume 58A, number 7
PHYSICS LETTERS
18 October 1976
SMALL-ANGLE SCATTERING OF X-RAYS BY ELECTRON-iRRADIATED COPPER A. SEEGER Max-Planck-Institut fur Metallforschung, Institut fur Physik, and Inst itut frr theoretische und angewandte Physik der Unir’ersitat Stuttgart, Stuttgart, Germany Received 6 August 1976
Small-angle scattering of annealed electron-irradiated copper shows that isolated monovacancies persist through recovery stage III. The scattering appearing at small angles after annealing between 260 K and 300 K can be accounted for by the condensation of seif-interstitials to form small dislocation loops.
Haubold [1] has recently reported measurements of the recovery of the diffuse scattering of X-rays by a copper single crystal that had been irradiated with 3 MeV electrons to an increase in residual resistivity of ~p0 130 n~lcm. The present note deals with the interpretation of the scattering observed at small angles after annealing to temperatures between 200 K and 300 K. This temperature range (known as recovcry stage III) is particularly interesting since competing models on the interpretation of radiation damage recovery [2, 3] differ in the nature of the point defects migrating in stage III. The striking feature after anneals at temperatures between 80 K and 200 K is the observation of an angular-independent (after correction for the atomic scattering intensity Ua) small-angle scattering 0(K) (approximately in (100)-direction) over a range of wavenumbers ~ extending from about 0.007 to 0.27 of the distance to the Brillouin zone boundary. This is attributed to the Laue scattering [4] of (essentially randomly distributed) vacant lattice sites. Haubold [1] shows that the absolute intensity comes out about right if the vacancy concentration C is calculated from the residual resistivity, assuming an average value of ~FP = 2 X l0~ ~2cm vacancy plus interstitial (which may be isolated, in a small cluster, or in a dislocation loop). During annealing between 260 K to 300 K a typical “small-angle scattering” (i.e., a scattering falling off rapidly with increasing distance from the centre of the first Brillouin zone) superimposed on an essentially constant background scattering appears. When normalized according to the remaining residual dcc-
trical resistivity, this background intensity is about the same as before the annealing. In order to be able to judge the physical significance of the results summarized above we have to consider the experimental errors of the determination of the scattering due to the radiation-induced defects. This intensity is obtained as the small difference of measurements (at 4 K) on an irradiated (an partially annealed) and on an unirradiated (dummy) specimen. An estimate of the uncertainty of the results may be deduced from the scatter of the intensity after the 200 K anneal. The experimental points are compatible with Haubold’s interpretation of scattering by a random distribution of vacancies if an error in (U/COa)200 K of ±0.25 is admitted. The band (u/Ccra)200 K = 0.75 ±0.25 containing all experimental points is indicated in the upper part of fig. 1 by dense hatching. Annealing at 300 K reduces the background intensity relative to that after 200 K annealing by about a factor 2.5. For an individual measurement the absolute error of a should remain approximately the same, since it results from subtracting two approximately equal quantities. This means that in going from 200 K to 300 K the relative error increases approximately by a factor of 2.5. Light hatching in the upper part of fig. 1 indicates the error band 0 25 X 2 5 (U/CUa)300K = 0.75 ± 1 2 = 0.75 ±0.4. (1) 2 / The factor 21~’2takes into account that the data points after the 300 K anneal are averages of two sirnilar measurements (H.G. Haubold, personal1 communiall points cation). It will be seen that for i< > 5 nm— 481 -
Volume 58A, number 7
/
PHYSICS LETTERS
annealing temperature 200K: 300K:
.
0 +
+
18 Oetober 1976
Analysis of Guinier plot: and comparison with theory ~
Anneal.
R
temp. (K)
(nm)
5
+
Exp. +
~ 0
‘
‘
-
‘
p0/b
o(O)/Coa Calc.
--------—--~-~-~—-~‘
‘
-‘
..-
‘
.-
.‘
0.75
—
280 3
0.44
ft60
n
--
5 7.212
6
23
13
//////////// -7
I
I
I
76
36
64
I
4
~ 2)
(100)
(nm
\,
280 K: • 300 K: +
_________________________
0
_________________________________
04
76
36
2
64 (nm ~)
predicts that the background intensity should have disappeared after 300 K the annealing. Let us now consider origin of the intensity increase at small scattering angles. Subtracting the background intensity according to (1) gives us the lower part of fig. 1. From the slope of the resulting “Guinier plot” (dashed lines) we may deduce the “radius of gyration” R~[4] of the defects causing the smallangle scattering. Table I gives R 5 and the intensity cxtrapolated to K = 0, a(0), both after the 300 K anneal and after a 280 K anneal. As realized by Haubold [1] o(0) is much too small to be attributed to the scattering formed the vacancies present after the 200 of K voids anneal. In an by attempt to remain compatible
Fig. 1. Guinier plot of scattered intensity with (lower part) and without (upper part) subtraction of wavenumber independent scattering,
with Schilling’s model 31 he proposes that the scattering is due to flat platelets of vacancies formed by vacancy migration. However, it remains unexplained
(measured after 300 K anneal) fall inside the error band (1). We have to draw the conclusion that after 300 K annealing a constant diffuse scattering background exists that, within experimental error, has about the same intensity-to-residual-resistivity ratio as after the 200 K anneal, Since the only conceivable origin of the background intensity are isolated vacancies, we conclude that all that happens to monovacancies in stage III is that their concentration is reduced. This is in full agreement with one of the above-mentioned models [2], according to which the decrease of the electrical resistivity in stage Ill is mainly due to the annihilation of virtually immobile vacancies by seif-interstitials migrating towards them. The competing model [3] postulates that monovacancies disappear in stage III by migration, either annihilating at preexisting interstitial clusters or at trapped interstitials, or forming vacancy clusters. It is obvious that the experimental results [1] are incompatible with this model, since it
why such flat platelets should be stable at room ternperature and why they should not collapse to form dislocation rings. An equally strong objection against this interpretation is that it is not at all clear why some monovacancies should form clusters while the rest remains dispersed as evidenced by the existence of the background intensity after the 300 K anneal. In the other model [2] a natural interpretation of the small-angle scattering is that self-interstitials migrating in stage III have formed small dislocation loops. The small-angle scattering from dislocation loops has been treated in a number of papers [5—8]. For loops giving rise to radii of gyration as small as those listed in table 1 the treatment by the linear theory of elasticity does not suffice; one has to inelude at least effects quadratic in the strains [8]. Unfortunately, most of the explicit results on quadratic effects on circular edge-dislocation loops in copper [8] pertain to ioops formed by condensation of vacancies. Nevertheless, the main characteristics of the scattering from interstitial loops may be deduced
482
Volume 58A, number 7
PHYSICS LETTERS
from the scattering amplitudes given separately for 1) and “quadratic” dilations “linear” dilations (a( (a(2)) [8]. We consider interstitial-type (pure edge) circular dislocation loops of strength b = 3~’2a 0 = 2.08 A, where a0 is the edge length of the elementary cube. The number of interstitials condensed in such a ring of radius p0 is =
4~ I~~0 \2 -___(~—_) ,
(2)
The enhancement of p0/R5 by the quadratic effects (fig. 5 of [7]) is smaller than for vacancy loops and leads to the radii given in table 1. Essentially the same radii are obtained from an estimate of the K-values at which the scattering intensity becomes undetectably 1~ and the a~2~ amplismall. interference of the a~ tudes isThe such that the intensity u(0) is enhanced relative to that calculated for the vacancy case. The enhancement increases with decreasing p 0 - We may write
2
2 4
0(0)b P0 33/2~\ ~ (P0~ (‘Ga n~22 b j
‘
‘
18 October 1976
demonstration [9, 10] that, at least over the size range accessible to transmission electron microscopy, Vacancy-type dislocation loops are not formed during stage-Ill annealing. It will be shown elsewhere that the trapping of positrons by the interstitial-type dislocation loops discussed above accounts well for the changes in the positron annihilation characteristics of electron-irradiated copper during stage-Ill annealing [11] . The interpretation given by Mantl and Triftsh~iuser[11] in terms of three-dimensional voids is incompatible with the experimental results of Haubold [1] since, as mentioned above, the condensation of the vacancies in voids should give rise to a much stronger small-angle X-ray scattering than observed. The comparison with positron trapping by dislocation loops in infavour neutron-irradiated copper [11] is no argument of the Mantl-—Triftsh~userinterpretation, since the interstitial loops considered in the present note are much smaller (n by at least a factor of hundred) than those observed in copper neutronirradiated above room temperature. The author acknowledges interesting discussions with Dr. H.G. Haubold.
where a
2.2 for p 0/b = 3.3 and a
2.6 for p0/b
2.3. This gives the theoretical values of table 1. We see that the calculated intensities do well account for the observed increase of the intensity of small angles, =
even if one takes into consideration that a fraction
[1] HG. Haubold, in: Fundamental aspects of radiation damage in metals, eds. M.T. Robinson and F.W. Young, U.S. ERDA CONF-751006-Pl, p. 268.
of the interstitials produced by the irradiation has condensed in loops too large to contribute to the scattering intensity [9, 10] We may summarize the conclusions of the present paper as follows: (i) In electron-irradiated Cu randomly distributed vacancies persist above 300 K. This may well be one of the most direct proofs that in Cu monovacancies
121 A. Seeger, in: Fundamental aspects of radiation damage in metals eds. MT. Robinson and F.W. Young, ERDA CONF-75 1006, P1, p.493. 13] W. Schilling, P. Ehrhart and K. Sonnenberg, Fundamental aspects of radiation damage in metals, eds. M.T. Robinson and F.W. Young, U.S. ERDA CONF-75 1006P1, p. 470. 141 A. Guinier and F. Fournet, Small-angle scattering of Xrays (Wiley, New York 1955). [5] A. Seeger and E. Kr6ner, Z. Naturforschg. 14a (1959) 74.
are immobile at 300 K. (ii) The small-angle scattering observed after annealing between 260 K and 300 K can be attributed to the formation of small dislocation 1oops by selfinterstitials migrating in annealing stage III. Since the
[6] A. Seeger and M. RUhle, Z. Physik 11(1963)216. [7] A. Seeger, V. Gerold and M. Riihle, Z. Metallkunde 54 (1963) 493. [8] A. Seeger and P. Brand, in: Small-angle X-ray scattering, ed. H. Brumberger (Gordon and Breach, New York, 1967) p. 383.
application of [81 to the present problem is preliminary, it is desirable to carry out a more complete calculation based on anisotropic non-linear elasticity theor The preceding conclusions are in accord with a large body of evidence [2] .Here we mention the
191 S. Ohr, in: Fundamental aspects of radiation damage in metals, eds. MT. Robinson and F.W. Young, U.S. ERDA CONF-751006-P1, p.650. [10] W. J~gerand K. Urban, submitted to Commun. Phys. [11] S. Mantl and W. Triftshäuser, in: Fundamental aspects of radiation damage in metals, eds. M.T. Robinson and F.W. Young, U.S. ERDACONF-751006-P2,p. 1122.
483
PHYSICS LETTERS
18 October 1976
SMALL-ANGLE SCATTERING OF X-RAYS BY ELECTRON-iRRADIATED COPPER A. SEEGER Max-Planck-Institut fur Metallforschung, Institut fur Physik, and Inst itut frr theoretische und angewandte Physik der Unir’ersitat Stuttgart, Stuttgart, Germany Received 6 August 1976
Small-angle scattering of annealed electron-irradiated copper shows that isolated monovacancies persist through recovery stage III. The scattering appearing at small angles after annealing between 260 K and 300 K can be accounted for by the condensation of seif-interstitials to form small dislocation loops.
Haubold [1] has recently reported measurements of the recovery of the diffuse scattering of X-rays by a copper single crystal that had been irradiated with 3 MeV electrons to an increase in residual resistivity of ~p0 130 n~lcm. The present note deals with the interpretation of the scattering observed at small angles after annealing to temperatures between 200 K and 300 K. This temperature range (known as recovcry stage III) is particularly interesting since competing models on the interpretation of radiation damage recovery [2, 3] differ in the nature of the point defects migrating in stage III. The striking feature after anneals at temperatures between 80 K and 200 K is the observation of an angular-independent (after correction for the atomic scattering intensity Ua) small-angle scattering 0(K) (approximately in (100)-direction) over a range of wavenumbers ~ extending from about 0.007 to 0.27 of the distance to the Brillouin zone boundary. This is attributed to the Laue scattering [4] of (essentially randomly distributed) vacant lattice sites. Haubold [1] shows that the absolute intensity comes out about right if the vacancy concentration C is calculated from the residual resistivity, assuming an average value of ~FP = 2 X l0~ ~2cm vacancy plus interstitial (which may be isolated, in a small cluster, or in a dislocation loop). During annealing between 260 K to 300 K a typical “small-angle scattering” (i.e., a scattering falling off rapidly with increasing distance from the centre of the first Brillouin zone) superimposed on an essentially constant background scattering appears. When normalized according to the remaining residual dcc-
trical resistivity, this background intensity is about the same as before the annealing. In order to be able to judge the physical significance of the results summarized above we have to consider the experimental errors of the determination of the scattering due to the radiation-induced defects. This intensity is obtained as the small difference of measurements (at 4 K) on an irradiated (an partially annealed) and on an unirradiated (dummy) specimen. An estimate of the uncertainty of the results may be deduced from the scatter of the intensity after the 200 K anneal. The experimental points are compatible with Haubold’s interpretation of scattering by a random distribution of vacancies if an error in (U/COa)200 K of ±0.25 is admitted. The band (u/Ccra)200 K = 0.75 ±0.25 containing all experimental points is indicated in the upper part of fig. 1 by dense hatching. Annealing at 300 K reduces the background intensity relative to that after 200 K annealing by about a factor 2.5. For an individual measurement the absolute error of a should remain approximately the same, since it results from subtracting two approximately equal quantities. This means that in going from 200 K to 300 K the relative error increases approximately by a factor of 2.5. Light hatching in the upper part of fig. 1 indicates the error band 0 25 X 2 5 (U/CUa)300K = 0.75 ± 1 2 = 0.75 ±0.4. (1) 2 / The factor 21~’2takes into account that the data points after the 300 K anneal are averages of two sirnilar measurements (H.G. Haubold, personal1 communiall points cation). It will be seen that for i< > 5 nm— 481 -
Volume 58A, number 7
/
PHYSICS LETTERS
annealing temperature 200K: 300K:
.
0 +
+
18 Oetober 1976
Analysis of Guinier plot: and comparison with theory ~
Anneal.
R
temp. (K)
(nm)
5
+
Exp. +
~ 0
‘
‘
-
‘
p0/b
o(O)/Coa Calc.
--------—--~-~-~—-~‘
‘
-‘
..-
‘
.-
.‘
0.75
—
280 3
0.44
ft60
n
--
5 7.212
6
23
13
//////////// -7
I
I
I
76
36
64
I
4
~ 2)
(100)
(nm
\,
280 K: • 300 K: +
_________________________
0
_________________________________
04
76
36
2
64 (nm ~)
predicts that the background intensity should have disappeared after 300 K the annealing. Let us now consider origin of the intensity increase at small scattering angles. Subtracting the background intensity according to (1) gives us the lower part of fig. 1. From the slope of the resulting “Guinier plot” (dashed lines) we may deduce the “radius of gyration” R~[4] of the defects causing the smallangle scattering. Table I gives R 5 and the intensity cxtrapolated to K = 0, a(0), both after the 300 K anneal and after a 280 K anneal. As realized by Haubold [1] o(0) is much too small to be attributed to the scattering formed the vacancies present after the 200 of K voids anneal. In an by attempt to remain compatible
Fig. 1. Guinier plot of scattered intensity with (lower part) and without (upper part) subtraction of wavenumber independent scattering,
with Schilling’s model 31 he proposes that the scattering is due to flat platelets of vacancies formed by vacancy migration. However, it remains unexplained
(measured after 300 K anneal) fall inside the error band (1). We have to draw the conclusion that after 300 K annealing a constant diffuse scattering background exists that, within experimental error, has about the same intensity-to-residual-resistivity ratio as after the 200 K anneal, Since the only conceivable origin of the background intensity are isolated vacancies, we conclude that all that happens to monovacancies in stage III is that their concentration is reduced. This is in full agreement with one of the above-mentioned models [2], according to which the decrease of the electrical resistivity in stage Ill is mainly due to the annihilation of virtually immobile vacancies by seif-interstitials migrating towards them. The competing model [3] postulates that monovacancies disappear in stage III by migration, either annihilating at preexisting interstitial clusters or at trapped interstitials, or forming vacancy clusters. It is obvious that the experimental results [1] are incompatible with this model, since it
why such flat platelets should be stable at room ternperature and why they should not collapse to form dislocation rings. An equally strong objection against this interpretation is that it is not at all clear why some monovacancies should form clusters while the rest remains dispersed as evidenced by the existence of the background intensity after the 300 K anneal. In the other model [2] a natural interpretation of the small-angle scattering is that self-interstitials migrating in stage III have formed small dislocation loops. The small-angle scattering from dislocation loops has been treated in a number of papers [5—8]. For loops giving rise to radii of gyration as small as those listed in table 1 the treatment by the linear theory of elasticity does not suffice; one has to inelude at least effects quadratic in the strains [8]. Unfortunately, most of the explicit results on quadratic effects on circular edge-dislocation loops in copper [8] pertain to ioops formed by condensation of vacancies. Nevertheless, the main characteristics of the scattering from interstitial loops may be deduced
482
Volume 58A, number 7
PHYSICS LETTERS
from the scattering amplitudes given separately for 1) and “quadratic” dilations “linear” dilations (a( (a(2)) [8]. We consider interstitial-type (pure edge) circular dislocation loops of strength b = 3~’2a 0 = 2.08 A, where a0 is the edge length of the elementary cube. The number of interstitials condensed in such a ring of radius p0 is =
4~ I~~0 \2 -___(~—_) ,
(2)
The enhancement of p0/R5 by the quadratic effects (fig. 5 of [7]) is smaller than for vacancy loops and leads to the radii given in table 1. Essentially the same radii are obtained from an estimate of the K-values at which the scattering intensity becomes undetectably 1~ and the a~2~ amplismall. interference of the a~ tudes isThe such that the intensity u(0) is enhanced relative to that calculated for the vacancy case. The enhancement increases with decreasing p 0 - We may write
2
2 4
0(0)b P0 33/2~\ ~ (P0~ (‘Ga n~22 b j
‘
‘
18 October 1976
demonstration [9, 10] that, at least over the size range accessible to transmission electron microscopy, Vacancy-type dislocation loops are not formed during stage-Ill annealing. It will be shown elsewhere that the trapping of positrons by the interstitial-type dislocation loops discussed above accounts well for the changes in the positron annihilation characteristics of electron-irradiated copper during stage-Ill annealing [11] . The interpretation given by Mantl and Triftsh~iuser[11] in terms of three-dimensional voids is incompatible with the experimental results of Haubold [1] since, as mentioned above, the condensation of the vacancies in voids should give rise to a much stronger small-angle X-ray scattering than observed. The comparison with positron trapping by dislocation loops in infavour neutron-irradiated copper [11] is no argument of the Mantl-—Triftsh~userinterpretation, since the interstitial loops considered in the present note are much smaller (n by at least a factor of hundred) than those observed in copper neutronirradiated above room temperature. The author acknowledges interesting discussions with Dr. H.G. Haubold.
where a
2.2 for p 0/b = 3.3 and a
2.6 for p0/b
2.3. This gives the theoretical values of table 1. We see that the calculated intensities do well account for the observed increase of the intensity of small angles, =
even if one takes into consideration that a fraction
[1] HG. Haubold, in: Fundamental aspects of radiation damage in metals, eds. M.T. Robinson and F.W. Young, U.S. ERDA CONF-751006-Pl, p. 268.
of the interstitials produced by the irradiation has condensed in loops too large to contribute to the scattering intensity [9, 10] We may summarize the conclusions of the present paper as follows: (i) In electron-irradiated Cu randomly distributed vacancies persist above 300 K. This may well be one of the most direct proofs that in Cu monovacancies
121 A. Seeger, in: Fundamental aspects of radiation damage in metals eds. MT. Robinson and F.W. Young, ERDA CONF-75 1006, P1, p.493. 13] W. Schilling, P. Ehrhart and K. Sonnenberg, Fundamental aspects of radiation damage in metals, eds. M.T. Robinson and F.W. Young, U.S. ERDA CONF-75 1006P1, p. 470. 141 A. Guinier and F. Fournet, Small-angle scattering of Xrays (Wiley, New York 1955). [5] A. Seeger and E. Kr6ner, Z. Naturforschg. 14a (1959) 74.
are immobile at 300 K. (ii) The small-angle scattering observed after annealing between 260 K and 300 K can be attributed to the formation of small dislocation 1oops by selfinterstitials migrating in annealing stage III. Since the
[6] A. Seeger and M. RUhle, Z. Physik 11(1963)216. [7] A. Seeger, V. Gerold and M. Riihle, Z. Metallkunde 54 (1963) 493. [8] A. Seeger and P. Brand, in: Small-angle X-ray scattering, ed. H. Brumberger (Gordon and Breach, New York, 1967) p. 383.
application of [81 to the present problem is preliminary, it is desirable to carry out a more complete calculation based on anisotropic non-linear elasticity theor The preceding conclusions are in accord with a large body of evidence [2] .Here we mention the
191 S. Ohr, in: Fundamental aspects of radiation damage in metals, eds. MT. Robinson and F.W. Young, U.S. ERDA CONF-751006-P1, p.650. [10] W. J~gerand K. Urban, submitted to Commun. Phys. [11] S. Mantl and W. Triftshäuser, in: Fundamental aspects of radiation damage in metals, eds. M.T. Robinson and F.W. Young, U.S. ERDACONF-751006-P2,p. 1122.
483