Nuclear Instruments and Methods in Physics Research B 240 (2005) 719–725 www.elsevier.com/locate/nimb

Determination of absolute defect concentrations for saturated positron trapping – deformed polycrystalline Ni as a case study R. Krause-Rehberg

a,*

, V. Bondarenko a, E. Thiele b, R. Klemm c, N. Schell

d

a Universita¨t Halle, FB Physik, Fr. Bach Platz 6, 06099 Halle (Saale), Germany TU Dresden, Inst. fu¨r Angewandte Physik/Halbleiterphysik, 01062 Dresden, Germany c TU Dresden, Inst. fu¨r Strukturphysik, 01062 Dresden, Germany Inst. fu¨r Ionenstrahlphysik und Materialforschung, FZ Rossendorf, 01314 Dresden, Germany b

d

Received 5 May 2004; received in revised form 20 April 2005 Available online 5 July 2005

Abstract Positrons may be used in many cases to determine defect densities of vacancies and dislocations. In case of saturated positron trapping, i.e. all positrons are getting trapped, only a lower-limit estimation can be given. However, a combination of positron back-diffusion measurements using a monoenergetic positron beam in combination with conventional lifetime spectroscopy can be used to overcome the problem of saturated positron trapping. As a case study, this combination was used for the determination of dislocation densities in polycrystalline nickel samples of highly varying dislocation density. Saturated positron trapping into dislocations and small voids was observed. The total positron trapping rate was calculated from the positron diffusion length obtained by back-diffusion experiments. The trapping rates of the two defects were finally obtained using the decomposition of lifetime spectra. The results were found in good agreement with those determined by the analysis of synchrotron Bragg-diffraction profiles, measured on the same set of samples. From the comparison of both techniques, the positron trapping coefficient was found to be ldisl = 3.9 ± 0.3 cm2/s for a high density of dislocations in Ni.
2005 Elsevier B.V. All rights reserved. PACS: 78.70.Bj Keywords: Positron annihilation; VEPAS; Dislocation density; Nickel; Specific trapping rate; Bragg-diffraction profiles

*

Corresponding author. Tel.: +49 345 5525567; fax: +49 345 5527160. E-mail address: [email protected] (R. Krause-Rehberg).

0168-583X/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.04.130

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R. Krause-Rehberg et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 719–725

1. Introduction Positron annihilation is a powerful technique for the identification of open-volume-type defects such as vacancies, small vacancy clusters, and dislocations [1]. Because the positron lifetime is sensitive to the electron density at the trapping site, positron lifetime spectroscopy (PALS) allows the determination of the open-volume size of the defect. In addition, Doppler broadening coincidence spectroscopy provides information on the chemical environment of the defect. Thus, the nature of positron-active defects can be studied in detail. Moreover, an increasing defect density gives rise to a larger number of positrons being trapped by the defects. The trapping rate j, which is experimentally determined increases. The trapping rate is directly linked to the defect density qdef. In many cases, as for vacancies and dislocations, j is proportional to qdef. The proportionality constant l is called trapping coefficient (Eq. (2)). It must be determined by a reference method. For a high defect density, all positrons will be trapped and almost no positrons annihilate in the defect-free bulk of the sample. This situation is called saturation trapping. Then, the positron lifetime being typical for the defect (or defects) is measured and only a lower limit estimation can be given for the defect density. However, back-diffusion measurements of a monoenergetic positron beam (VEPAS – variable energy positron annihilation spectroscopy) have a higher sensitivity limit, i.e. positrons implanted with small energy in the keV range close to the surface have still a high probability to reach the surface when the defect density is in the saturation range for PALS. From the positron diffusion data the total trapping rate can be derived. However, this technique is not at all sensitive for the defect type. A combination of both techniques, back-diffusion measurements and PALS may give information on the defects type (or types) and the individual densities. In the presented paper this combination was utilized to study plastically deformed Ni samples showing densities of dislocations, high enough for saturation trapping. The total trapping rate was determined by back-diffusion measurements and the

fraction of positrons being trapped in dislocations was derived from the lifetime spectra decomposition. Moreover, the dislocation density was determined by synchrotron radiation Braggdiffraction as a reference method for the whole set of samples, so that the trapping coefficient for dislocations in Ni for high dislocation densities can be determined.

2. Samples The various experimental investigations presented here were performed on polycrystalline nickel samples of four different material classes produced by special methods. These material classes differ from each other by the grain size, by the type and density of lattice defects as well as by the content of impurities. For all the classes samples were analysed in the initial stage after production and, with one exception, after cyclic plastic deformation at room temperature, too. The cyclic loading was carried out with constant plastic strain amplitudes epa up to a cycle number where a mechanically stabilized state was reached. It is well accepted that at the given deformation temperature and for the range of grain size examined the plastic deformation is essentially caused by dislocation processes. The sample of the first material class (Ni1) was prepared from ball milled nickel powder by a compaction at 600 C for 1 h at a pressure of 480 MPa followed by a special multi-forging compaction procedure at room temperature (BMCMF) leading to a mass density of 98.5% of the theoretical value, which was not sufficient for the performance of a cyclic plastic deformation. The purity of the Ni material was 99.2 at.% (0.38 at.% oxygen, 0.27 at.% iron). The specimens of the second (Ni2) and third (Ni3) material class were obtained from Ni (purity 99.992 at.%) subjected to a special type of severe plastic deformation called equal channel angular pressing which was carried out at room temperature (RTECAP) and at 250 C (ETECAP), respectively [2,3]. The grains in these materials were slightly coarsened due to the cyclic plastic deformation.

R. Krause-Rehberg et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 719–725

Samples of the fourth material class (Ni4) were produced from RTECAP-Ni (purity 99.992 at.%) by a heat treatment at 1230 K for 2 min which yields to a recrystallization (R) of the material. The grain structure of the samples is stable towards cyclic plastic deformation [4]. Table 1 contains details of the sample designation due to their production method, of the sample deformation stage as well as of the mean grain size d of the structure. The experimental method for the determination of d is added, too, because the amount of d depends on it in a characteristic manner [5]. As an example, in Fig. 1, micrographs from bright field transmission electron microscopy (TEM) are given showing the defect structure in

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the cyclically deformed samples of different materials classes. Obviously, in consequence of the cyclic plastic deformation typical dislocation structures were developed mainly consisting of dislocation rich walls, surrounding dislocation poor regions or of dislocations fitted to the grain boundaries. The walls are thinner for higher deformation amplitudes. Besides, remnants of the initial defect structure can be observed, as for example in Fig. 1(a), where extended regions of high dislocation content are left. Both, the Bragg-diffraction as well as the positron techniques will give an average defect density integrated over the penetration depth of the synchrotron radiation and of the positrons (for PALS several 10 lm).

Table 1 Designation, production method, deformation stage and mean grain size d of the samples investigated Sample designation

Production method

Deformation stage

Mean grain size d (nm)

qdisl (cm
2)

Ni1-0 Ni2-0 Ni2-1 Ni2-2 Ni3-0 Ni3-1 Ni3-2 Ni4-0 Ni4-1 Ni4-2

BMCMF RTECAP RTECAP RTECAP ETECAP ETECAP ETECAP R R R

Initial Initial epa = 2.5 · 10
4 epa = 2.0 · 10
3 Initial epa = 2.5 · 10
4 epa = 2.0 · 10
3 Initial epa = 2.5 · 10
4 epa = 2.0 · 10
3

50 500 600 700 800 2000 2000 2500 2500 2500

9.3 · 1010 1.0 · 1011 3.2 · 1010 2.4 · 1010 6.2 · 1010 2.7 · 1010 1.6 · 1010 1.0 · 109 6.0 · 109 8.0 · 109

The grain size of sample Ni1-0 was determined by synchrotron radiation diffraction analysis [2]. All other samples were investigated by electron back scattering diffraction technique (EBSD) in a scanning electron microscope with field emission gun [3]. qdisl is the dislocation density as obtained by the analysis of synchrotron Bragg-diffraction profiles.

Fig. 1. TEM bright field images for sample Ni2-2 (a), Ni3-2 (b) and Ni4-2 (c), all of them cyclically deformed at epa = 2.0 · 10
3.

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3. Experimental techniques 3.1. Positron annihilation In our samples, always two-component lifetime spectra were found showing two lifetimes larger than the bulk value in Ni (sb = 109 ps). Thus, saturated positron trapping was observed. The lifetime components were attributed to the trapping by dislocations (s1 = 150–180 ps) and small voids (s2 = 350–400 ps). Since almost all positrons are trapped, the intensities of these components I1 and I2 cannot be used to calculate the trapping rate and the defect density of the two defects. However, the intensities give the fraction of the total trapping rate jtotal originating from the two types of defects: jtotal ¼ jdisl þ jvoids ; jdisl ¼ I 1 jtotal ; jvoids ¼ I 2 jtotal ; I 1 þ I 2 ¼ 1.

ð1Þ

For trapping by dislocations, a direct proportionality between the trapping rate jdisl and the dislocation density qdisl is expected [6]. The trapping coefficient ldisl was earlier determined for weakly deformed Ni to be ldisl = 1.2 cm2/s [7–9]. However, in case of a high deformation degree this constant may be different. jdisl ¼ ldisl  qdisl .

ð2Þ

The total trapping rate jtotal can be obtained from the positron diffusion length L which is obtained by a back-diffusion experiment using a monoenergetic positron beam [1] (see discussion of Fig. 4). Lref is the diffusion length measured in a well annealed, defect-free Ni single crystal (Lref = 116.5 nm). !
2 1 Lref jtotal ¼
1 . ð3Þ sb L Combining Eqs. (1)–(3) one obtains the dislocation density as results of the parameters of both positron experiments; I1 from PALS and L from the back-diffusion measurement of VEPAS. !
2 I1 Lref qdisl ¼
1 . ð4Þ ldisl sb L

In our case the dislocation density is known from a reference method. Thus, Eq. (4) can be used to calculate the trapping coefficient ldisl for dislocations in case of saturation trapping conditions. However, it should be noted that this procedure can only be applied when the defect density is expected to be constant in the near-surface sensitivity range (1 lm) of the positron beam technique (determination of L) and the information depth of positron lifetime (a few 10 lm). This is to be expected here. 3.2. Bragg-diffraction with synchrotron radiation The Bragg-diffraction experiments were carried out at the beamline BM 20 (ROBL) of the European Synchrotron Radiation Facility in Grenoble. A linear polarized, collimated and vertical focused synchrotron radiation was used which was monochromized at a wavelength of k = 0.1534 nm with an energy resolution of 1.5 · 10
4. The cross-section of the incident beam was 0.2 · 2 mm2. Bragg-diffraction profiles of {1 1 1}, {2 0 0}, {2 2 0}, {3 1 1}, {2 2 2}, {4 0 0} and {3 3 1} reflection type were measured with a position sensitive detector, adjusted at the seven corresponding mean Bragg angles 2#. It was possible to exploit the signal of 3380 channels for a D2#-interval of 8.28 at each 2#-position. For all measurements the diffraction vectors g were parallel to the sample surface normal z which was perpendicular to the loading axis in the case of the cyclically deformed samples. In order to ensure that a sufficient number of crystallites is in Bragg-position also for the coarse-grained Ni, the samples were rotated round z during the measurements to enlarge the irradiated volume. To enable the determination of the total dislocation density q by the method of Krivoglaz–Wilkens [10], at first a background correction and a smoothing of the measured Bragg-diffraction profiles with negligible instrumental broadening were done. Then, for the data evaluation the experimental D2#-intervals were cut to achieve a common minimum Fourier length a3 = 4 nm for all reflection types. The Fourier coefficients AL of the profiles with n as the Fourier order and L = na3 as the Fourier length were analytically calculated as usual.

R. Krause-Rehberg et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 719–725

In principle it is possible to get the q-value for a sample from the information content of only one single diffraction profile applying the Krivoglaz– Wilkens algorithm. Assuming that solely dislocations are responsible for the internal strains in the sample, the size dKW of coherently scattering regions, the total dislocation density q and an outer cut-off radius Reff for the stress field of the dislocation arrangement can be determined from the function
 ln AL 1 Reff W¼
2 ¼ þ B ln ð5Þ Ld KW L L

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12

Ni1 BMCMF Ni2 RTECAP Ni3 ETECAP Ni4 R

10 8 6 4 2 0 0

5

10

15

20

25

2

with B ¼ p2 ðbgÞ Cq, where b is the Burgers vector, g is the diffraction vector and C is a mean contrast factor for the dislocations in an elastically isotropic consideration. The C values were calculated according to Wilkens [11] for the seven {h k l} reflection types investigated, assuming equal densities of screw and edge dislocations on all slip systems [2]. Mean values of dKW, q and Reff for one sample were obtained by a best-fitting procedure of the function W(ln L) for each profile and by averaging the results for the seven different profiles measured at the sample. In this way, the influence of the elastic anisotropy in Ni on the broadening of different {h k l} reflection type profiles and, therefore, on the value of q is compensated in a simpler manner as proposed for example by Unga´r et al. [12]. On the other hand, the calculated dislocation densities should be upper limits in general, because the contribution of long-range internal strains (e.g. granular strains) to the profile broadening is neglected.

4. Results and discussion In Fig. 2 the results are presented for the determination of the mean dislocation density q in dependence on the deformation stage of the samples. They were characterized by the plastic strain amplitude epa, where epa = 0 represents the as-produced state. For the Bragg-diffraction experiments the error Dq is the standard deviation of the q values calculated from the seven {h k l} reflection type profiles.

Fig. 2. Mean dislocation density qdisl for the Ni samples of the different material classes in dependence on the plastic strain amplitude epa (initial stage: epa = 0). q is determined from Bragg-diffraction measurements. For the sample classes Ni2 and Ni3 the results of some additionally investigated samples are included.

Considering the findings of the Bragg-diffraction experiments, as expected, the dislocation density is high for the initial stage of samples with the exception of the recrystallized sample Ni4-0. Furthermore, it is reasonable that q is higher in the sample Ni2-0 than in Ni3-0 because of the lower temperature during sample treatment which leads to a higher equilibrium dislocation density for the ECAP-process. The dislocation density of the sample Ni1-0 (BMCMF) was found to be between the values for the samples Ni2-0 and Ni3-0. In the range of plastic strain amplitudes investigated, the cyclic plastic deformation leads to a decrease of q with increasing epa in the case of Ni2 and Ni3. This should be a result of dislocation annihilation processes in consequence of a rearrangement of the initially stored dislocations due to the dislocation-governed plastic deformation. Contrary, for the recrystallized material Ni4 the dislocation density increases with epa. It seems that the development of the fatigue-induced dislocation wall structure, which is observed in all material classes, results in a nearly equal dislocation density at higher epa independent of the prehistory of the micro- and submicro-crystalline samples. Positron lifetime experiments and back-diffusion measurements using VEPAS were performed

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for the same set of samples investigated earlier by Bragg-diffraction using synchrotron radiation. Fig. 3 shows exemplarily four VEPAS profiles. The basics of the method and the physical meaning of the S and W parameters are explained in [1]. In brief, the positrons are implanted into the sample with increasing energy, i.e. depth. The S and W parameters reflect the electron momentum distribution which is different at the surface and in the bulk of the sample. Because trapping in the bulk is saturated (the total trapping rate is much larger than the annihilation rate in the bulk: jtotal  1/ sb), there are only three annihilation sites for the

positrons: the surface potential, dislocations (lifetime component of about 150 ps), and small vacancy clusters (350 ps). The latter two traps give rise for the saturated S and W parameters at high energies. The positrons cannot diffuse back to the surface from there because of the high number of trapping centres, and the diffusion length is as shorter as higher the defect density in the sample will be. Therefore, the positron diffusion length and hence, the total trapping rate jtotal can be determined by the analysis of the S(E) and W(E) profiles. It was found that the two L values were always very similar for the same sample, and thus, they were averaged to give L and jtotal presented in Table 2. This table also shows the intensity of the dislocation-related lifetime component I1. According to Eq. (1), the trapping rate for positron trapping by the dislocations was calculated. In addition, the dislocation density obtained by the analysis of synchrotron Bragg-diffraction profiles is added in the last column of Table 1. Fig. 4 shows the plot of the trapping rate jdisl versus qdisl and the fit according to Eq. (2). The slope of the straight line represents the trapping coefficient ldisl for the case of saturated positron trapping by dislocations, ldisl = 3.9 ± 0.3 cm2/s. This value is slightly larger than the old value of 1.2 cm2/s [7–9]. However, the dislocation density range is different in both cases. The earlier value

Table 2 Results of the positron experiments

Fig. 3. Some examples of back-diffusion measurements using a monoenergetic positron beam (VEPAS). Shown is the S and W parameter as a function on the positron incident energy. Both parameters were normalised to the bulk values of defect-free Ni reference. The solid lines are fits to the data using VEPFIT [13].

Sample

L [nm]

jtotal (s
1)

I1 (%)

jdisl (s
1)

Ni1-0 Ni2-0 Ni3-0 Ni2-1 Ni2-2 Ni3-2 Ni3-1 Ni4-2 Ni4-1 Ni4-0 SC

16.0 ± 4.1 17.3 ± 2.8 24.9 ± 3.2 25.1 ± 2.7 25.3 ± 2.7 27.3 ± 3.5 35.3 ± 3.9 38.3 ± 3.9 48.1 ± 4.5 116.2 ± 5.4 116.5 ± 4.9

4.94E+11 4.23E+11 2.00E+11 1.96E+11 1.93E+11 1.64E+11 9.41E+10 7.85E+10 4.64E+10 0 0

67.9 ± 1.1 98.4 ± 0.3 98.5 ± 0.2 96.0 ± 0.4 93.2 ± 0.4 89.9 ± 0.4 93.1 ± 1.8 85.2 ± 1.4 66.1 ± 3.9 0 0

3.3 · 1011 4.1 · 1011 1.9 · 1011 1.8 · 1011 1.7 · 1011 1.4 · 1011 8.2 · 1010 6.1 · 1010 2.6 · 1010 0 0

L is the positron diffusion length, jtotal is the total trapping rate obtained by the back-diffusion experiment, I1 is the intensity of the dislocation-related lifetime component of the two-component spectra, and jdisl is the positron trapping rate of dislocations. SC denotes the well annealed Ni single crystal.

Trapping rate κdisl (1010 s-1)

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40 2

µdisl = (3.9 ± 0.3) cm /s

30

20

10

0 0

2

4

6

8

10

Dislocation density ρdisl (1010 cm-2) Fig. 4. Correlation between the dislocation density qdisl as determined by synchrotron Bragg-diffraction measurements and the positron trapping rate jdisl. The solid line is the result of a linear fit according to Eq. (2). The slope of the line corresponds to the positron trapping coefficient ldisl.

was determined for the case of non-saturated positron trapping, i.e. at smaller densities. Our value is valid in the range of qdisl = 1010  1011 cm
2, thus, for saturated positron trapping. Although the data scatter around the fitted straight line, the agreement of the results of both methods, positrons and synchrotron radiation, is rather satisfactory bearing in mind that for the two experimental methods the physical origin of the obtained information used to calculate the dislocation density is completely different.

5. Summary It could be shown that synchrotron Bragg-diffraction and positron annihilation experiments as two totally independent methods yield the same proportion between the corresponding qdisl data for all the Ni samples investigated. Therefore, both methods are suitable to pursue the relative change

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of the dislocation density after plastic deformation for a wide dislocation density range. Our study of deformed Ni shows as a case study that the combination of positron back-diffusion and positron lifetime spectroscopy allows the accurate determination of defects densities by positron annihilation still in the case of saturated positron trapping, which is impossible for the conventional positron lifetime or Doppler broadening spectroscopy. Possible applications are those cases were a high density of positron traps exists, e.g. agehardenable Al alloys containing nano-precipitations or materials exposed to a very high irradiation dose.

References [1] R. Krause-Rehberg, H.S. Leipner, Positron Annihilation in Semiconductors, Springer-Verlag, Berlin, 1999, ISBN 3-540-64371-0. [2] E. Thiele, M. Hecker, N. Schell, Mater. Sci. Forum 321– 324 (2000) 598. [3] R. Klemm, E. Thiele, H. Baum, C. Holste, in: Proc. 8th Int. Fatigue Congress, Stockholm, Sweden, 2002 (Cradley Heath), p. 1609. [4] E. Thiele, C. Holste, R. Klemm, Z. Metallkde. 93 (2002) 730. [5] E. Thiele, R. Klemm, L. Hollang, C. Holste, N. Schell, H. Natter, R. Hempelmann, Mater. Sci. Eng. A 390 (2005) 42. [6] P. Hautoja¨rvi (Ed.), Positrons in Solids, Vol. 12, Springer, Berlin, 1979. [7] G. Dlubek, O. Bru¨mmer, N. Meyendorf, Appl. Phys. 13 (1977) 67. [8] G. Dlubek, O. Bru¨mmer, N. Meyendorf, Phys. Stat. Sol. A 53 (1979) K157. [9] G. Dlubek, O. Bru¨mmer, N. Meyendorf, P. Hautoja¨rvi, A. Vehanen, J. Yli-Kauppila, J. Phys. F: Metal Phys. 9 (1979) 1961. [10] D. Breuer, P. Klimanek, U. Mu¨hle, U. Martin, Z. Metallkde. 88 (1997) 680. [11] M. Wilkens, Phys. Stat. Sol. (a) 2 (1970) 359. [12] T. Unga´r, A. Borbe´ly, Appl. Phys. Lett. 69 (1996) 3173. [13] A. van Veen, H. Schut, M. Clement, J.M.M. de Nies, A. Krusemann, M.R. Ijpma, Appl. Surf. Sci. 85 (1995) 216.