fulltext

Positron lifetimes in Cu-based b-phase alloys

(1_{) Elektrika eta Elektronika Saila, Euskal Herriko Unibertsitatea}

644 p.k., 48080 Bilbo, Spain

(2_{) IFIMAT, Universidad Nacional del Centro de la Provincia de Buenos Aires}

Pinto 399, 7000 Tandil, Argentina

Comisión de Investigaciones Científicas de la Provincia de Buenos Aires Buenos Aires, Argentina

(ricevuto il 7 Maggio 1996; approvato il 19 Novembre 1996)

Summary. — Experimental and theoretical characterization of the positron lifetimes

for bulk and vacancy-type defects are considered in the ordered b-phase Cu-Zn-Al alloys. The general trend exhibits the same behaviour in both cases, in which the bulk positron lifetimes vary very little with alloy’s composition. The ordered structure can be described as two interpenetrating sublattices; however, within the theoretical approach, there is no preferential positron trapping at the monovacancies of one of the two sublattices. The calculated lifetimes of positrons trapped at monovacancies depend mainly on the Al content in the next near-neighbourhood. The lifetimes of positrons trapped at divacancies are 10–25 ps larger than the ones at monovacancies. The experimental vacancy-type defect lifetimes are in good agreement with the ones calculated for monovacancies.

PACS 71.55.Ak – Metals, semimetals, and alloys. PACS 78.70.Bj – Positron annihilation.

PACS 61.72.Ji – Point defects (vacancies, interstitials, color centers, etc.) and defect clusters.

1. – Introduction

Within a certain range of composition Cu-based alloys show a high-temperature stable b.c.c. structure. This high-temperature phase can be retained at lower temperature by means of a suitable thermal treatment. During the cooling process the alloy undergoes one or two ordering processes. The first from the b.c.c.-phase to the first-neighbour ordered phase B2 and the second from B2 to the second-neighbour ordered phase (L21 or DO3). At a lower temperature a thermoelastic martensite transformation could take place. This first-order transition is diffusionless and proceeds by a shear mechanism. In Cu-based alloys the martensitic transformation and related phenomena have been the subject of extensive research because of the

technologically important shape memory effects linked with the martensitic

transformation. In these systems, vacancies play an important role because of their influence on processes such as ordering, phase decomposition and martensitic stabilization.

Positron trapping at lattice defects in solids is a well-known phenomenon, easily observed through marked effects on annihilation characteristics, which is at the basis of current techniques of defect detection. In particular, positron annihilation techniques have demonstrated to be one of the most adequate tools to study point defects in metals and alloys. In the case of Cu-based alloys, positron annihilation spectroscopy has proved to be a very useful and effective technique for studying the behavior of quenched-in vacancies [1-8].

The aim of this work is to characterize, experimental and theoretically, the lifetime values corresponding to the bulk state and the vacancy-type defects (mono- and divacancies) of the b phase as a function of the composition of the alloy in order to allow a correct interpretation of the point defects experimental results obtained from quenched and thermal equilibrium testing.

2. – Experimental procedure

The following alloys: Cu-48 at.% Zn, Cu-38 at.% Zn-5.0 at.% Al, Cu-21.69 at.% Zn-13.15 at.% Al, Cu-18.52 at.% Zn-14.74 at.% Al, and Cu-16.14 at.% Zn-15.93 at.% Al were prepared by melting 5N pure metals in sealed quartz capsules under argon atmosphere. From them single crystals were grown by the Bridgman technique, also in sealed quartz under argon atmosphere. Since no weight loss occurred during the preparation and growth, the nominal composition can be considered to be the real one. For the five alloys, the nominal martensite transformation temperatures are below room temperature. The samples for the positron lifetime measurements were cut by a low-speed diamond saw from the single crystals having a diameter of about 10 mm and a thickness of 2 mm. The damaged surface was etched away with a solution of 50% HNO3 in water.

The positron lifetime was measured by a conventional fast-fast timing coincidence system. The resolution function was approximated by a Gaussian having a FWHM of 250 ps. A22_{Na source of about 30 mCi contained in an envelope made from a thin Kapton} foil (1.1 mgcm22_{) was sandwiched in between two samples. More experimental details} are described in ref. [6]. The lifetime spectra were analyzed with the PATFIT computer program [9]. After subtraction of the source contribution, a two-component decomposition of the lifetime spectra gives a satisfactory fit. All positron lifetime measurements were performed at room temperature.

3. – Experimental results

The lifetime corresponding to the bulk state (Bloch state) can be obtained by measuring free defects samples. However, it is sometimes difficult to ensure which is the true value of the positron lifetime in the perfect lattice, particularly in the cases in which slowly migrating point-defects are retained in the material. In such a case, an alternative method can be applied to characterize the bulk state; first, by introducing mobile defects by, for example, quenching from intermediate or high temperatures and then to measure the evolution of the positron lifetime parameters when quenched-in defects are migrating as a function of the isothermal aging time. In this case, reliable

results are obtained by means of the application of the equations derived from the two-state standard trapping model to the experimental results. In the present work, following the ideas suggested by Aldi et al. [10] we have used the relationship

l14 lbulk1 (lbulk2 ltrap)

Itrap 1 2Itrap

, (1)

where Itrap denotes the relative intensity of the component, associated with vacancy-type defects, having the lifetime value ttrap4 l21trap; l21bulk4 tbulkrepresents the bulk positron lifetime and l21

1 4 t1 is the first lifetime component obtained from the PATFIT analysis of the spectra. Obviously, eq. (1) cannot be used in the saturation trapping case in which all positrons are annihilated into vacancy-type defects that yields ItrapB 1. However, eq. (1) works properly, away from saturation trapping at defects as shown by the plot l1 vs. ItrapO( 1 2 Itrap) (see fig. 1). On the other hand, we have obtained the lifetime characteristic of the vacancy-type defects from the mean statistical value (over more than 30 measurements in each case) of the second lifetime component resulting from the analysis of the positron lifetime spectra for quenching and aging experimental data in the same way as shown in ref. [6, 7]. Following eq. (1) the extrapolation at origin of the linear least-square fit of the data presented in fig. 1 gives a value for the bulk lifetime. However, the application of eq. (1) to the slope of the fitted straight line together with the experimental vacancy lifetime gives a second bulk

Fig. 1. – Isothermal aging time figure of l₁vs. ItrapO( 1 2 Itrap) measured at room temperature in

Cu-16.4 at.% Zn-15.93 at.% Al after quenching from 493 K into water at room temperature. l1

corresponds to the inverse of the first lifetime component and Itrap is the intensity of the second

component. tbulk4 126 ps is the average bulk lifetime (shown in table I) of the extrapolation of the

fitted curve at origin and the lifetime value obtained from the slope of the fitted curve, the experimental monovacancy lifetime and eq. (1).

TABLE I. – Experimentally measured positron lifetimes. tbulk is the positron lifetime for the

perfect lattice and ttrap is the trapped positron lifetime in vacancy-type defects. eOa is the

electronic concentration.

Composition in at.% eOa tbulk(ps) ttrap(ps) References

48 Zn 1.48 126 61 170 61 this work 38.0 Zn-5.0 Al 1.48 130 63 180 61 this work 23.9 Zn-9.02 Al 1.42 128 62 — [1] 21.69 Zn-13.15 Al 1.48 129 63 181 61 this work 18.52 Zn-14.74 Al 1.48 131 63 181 62 this work 16.14 Zn-15.93 Al 1.48 127 61 180 62 this work 14 Zn-17 Al 1.48 126 64 177 65 180 61 [12] [24]

lifetime value. The average of the previous two lifetimes obtained from the fitting procedure results in the bulk lifetime values shown in table I. Table I also shows the positron lifetimes corresponding to vacancy-type defects.

4. – Theoretical calculation scheme

It is possible to describe the B2 or L21 superlattices by four interpenetrating face-centred cubic (f.c.c.) sublattices [11], as shown in fig. 2. The various possible atomic configurations are then expressed in terms of the occupation probabilities Pv

i of the i-th

components, on the four f.c.c. sublattices n 4I, II, III and IV. For B2-type order PiI4

PII

i , PiIII4 PiIV, and for L21-type order PiI4 PiII, PiIIIcPiIV. The occupation probability

satisfies the following relations:

!

!

!

where Ci is the average concentration of the i-th element.

For the alloy compositions used in this study, the occupation probability is under the next constraints [12]

PCuI 4 PCuII4 1 ; PAlIII4 PCuIV4 0 .

As sublattices I and II are occupied by Cu atoms only, in order to simplify the description of the superlattice, we call a sublattice the addition of sublattices I and II, and b sublattice the addition of positions III and IV (Cu in site III, Zn in sites III and IV, and Al only in site IV).

The present calculations for the positron states and annihilation in perfect and defected lattices of Cu-Zn-Al alloys have been performed using the superimposed atom model of Puska and Nieminen [13]. This method, based on the non–self-consistent superposition of free atoms, reproduces rather well in a perfect metal host the electron density and the open volume in the interstitial regions [14]. In the case of a perfect host these are indeed the most important aspects of the positron state calculations, since due to the strong repulsion from the ion cores, the positron wave function is mainly, but

Fig. 2. – The four interpenetrating f.c.c. sublattices for describing the B2- or L21-type

superstructure.

not completely, localized in the interstitial regions between the ions. However, the calculations may overestimate the positron annihilation rate with core electrons leading to bulk lifetimes that may be 10% shorter than the experimental ones [14, 15]. In the case of vacancies, due to the reduced importance of the core annihilation, model calculations made by the two-component density-functional theory [16, 17] for the positron localized in a metal vacancy [18, 19] show that the self-consistency of the electron structure changes the positron annihilation rate only slightly. Taking into account that we are mainly interested in the composition influence of the studied alloys over the positron states in perfect and defected lattices, we have used the superimposed atom method, which is computationally much less demanding than the simultaneous self-consistent calculation of the electron density and the positron state. Details of the calculation method have been given elsewhere [20]. The supercell used in the calculations is a cube containing 16 atoms in the case of a perfect lattice. In the case of a monovacancy and a divacancy it contains 127 and 126 atoms, respectively. In the calculations we have also calculated the contributions to the annihilation rate of the atoms of sublattices a and b separately.

5. – Theoretical results and discussion

Figure 3 shows the lifetime values calculated in the constructed Cu-Zn-Al structures. The theoretical lifetimes are lower than the experimental ones in less than 10%. Taking into account the calculations overestimation of the positron annihilation rate with core electrons, the experimental results are within the range of error of the theoretical results. However, they are all within 3 ps, in good agreement with the trend shown by the experimental bulk lifetimes in table I. It is observed that for fixed concentrations of Al the positron lifetime does not depend almost anything on Zn

Fig. 3. – Positron bulk lifetimes for the perfect lattices Cu-Zn-Al calculated in supercells of 16 atoms vs. the Zn content of the alloy for different Al content.

concentration increase (Cu concentration decrease) in the alloy. Moreover, fig. 3 clearly indicates that the main positron lifetime variation with composition is produced by Al addition.

Table II presents the localized positron state energy at different monovancies constructed taking away one atom on sublattice a. The calculated lifetimes for Cu monovacancies located in sites I and II, of sublattice a, indicate that in Cu vacancies the positron lifetime does not depend on the site they are located. The positron lifetime depends mainly on the number of Al atoms located in the nearest-neighbourhood of the vacancy. The increase of the vacancy lifetime with the Al content in the

nearest-neighbourhood of the vacancy is mainly due to the fact that the

nearest-neighbours of a sublattice a site are atoms of sublattice b, and the next nearest-neighbours are Cu atoms of sublattice a for all the different vacancy locations in sublattice a. So, due to the strong positron wave function localization at the vacancy the differences in the annihilation rates of the positron in the vacancies are mainly due to the differences in the nearest-neighbour atoms of sublattice b.

Calculations have also been performed on 48 monovacancies constructed in sublattice b, corresponding to the compositions shown in table II. The calculations indicate that the lifetime does not depend on the atom taken away to form the vacancy. It depends mainly on the overall number of Al atoms in the next nearest-neighbourhood of the vacancy. The Cu by Zn substitution in the nearest-neighbourhood of the vacancy makes only second-order corrections to the positron lifetime. The positron binding energies overlap the ones shown in table II and the lifetime change in sublattice b vacancies goes from 179 ps eV (0–2 Al atoms present in the next nearest-neighbourhood of the vacancy) to 189 ps (12–14 atoms (Al 35–40 at.%) in the next nearest-neighbourhood of the vacancy).

TABLEII. – Theoretical positron lifetimes tvand positron binding energy, Eb, for Cu vacancies

of sublattice a. eOa is the electronic concentration. Composition in at.% eOa Vacancy location tv(ps) Eb(eV) Vacancy’s nearest-neighbours 50 Zn-0 Al 1.50 site I 178 1.2 8Zn

25 Zn-12.5 Al 1.50 site I 179 1.2 2Cu 4Zn 2Al

25 Zn-12.5 Al 1.50 site II 179 1.2 2Cu 4Zn 2Al

18.75 Zn-6.25 Al 1.31 site I 178 1.2 4Cu 3Zn 1Al

18.75 Zn-6.25 Al 1.31 site II 178 1.2 4Cu 3Zn 1Al

18.75 Zn-12.5 Al 1.44 site I 179 1.2 3Cu 3Zn 2Al

18.75 Zn-12.5 Al 1.44 site II 179 1.2 3Cu 3Zn 2Al

18.75 Zn-18.75 Al 1.56 site I 182 1.3 2Cu 3Zn 3Al

18.75 Zn-18.75 Al 1.56 site II 182 1.3 2Cu 3Zn 3Al

18.75 Zn-25 Al 1.69 site I 185 1.4 1Cu 3Zn 4Al

18.75 Zn-25 Al 1.69 site II 185 1.4 1Cu 3Zn 4Al

12.5 Zn-18.75 Al 1.50 site I 181 1.3 3Cu 2Zn 3Al

12.5 Zn-18.75 Al 1.50 site II 181 1.3 3Cu 2Zn 3Al

Therefore, the calculations indicate that there is no preferential positron trapping in one of the two sublattices and the positron lifetime value varies in the studied vacancies depending on the Al content in the near neighbourhood by B12 ps. Taking into account that in the disordered and metastable b phase of the Cu-Zn-Al alloy at room temperature, Al-rich zones are not expected due to the large mobility of vacancies at such a temperature, the monovacancy positron lifetime predicted by the calculations is close to 180 ps, in good agreement with the experimental results presented in table I. Thermal-equilibrium Doppler broadening measurements performed in b.c.c. order-disorder alloy b Cu-Zn [21] show a small peak coincidence counts jump approaching the order-disorder transition temperature, attributed to the rapid formation of equilibrium divacancies. However, in opposition to this explanation Schultz and MacKenzie [22] obtain a slight narrowing of the symmetric lifetime peak below the order-disorder transition temperature in b Cu-Zn. Recent thermal-equilibrium lifetime measurements performed in b-phase Cu-Zn-Al alloys [23] show a small lifetime jump too, approaching the B2 order temperature. However, the spectra cannot be decomposed in another component due to the fact that the positron lifetime increase amounts to only 8 ps with respect to the positron lifetime corresponding to the saturation signal at monova-cancies. In order to clarify the divacancy contribution we have also performed theoretical calculations for divacancies. The divacancies have been taken by removing either two atoms located in directions [111] or in directions [100]. Table III indicates that the lifetimes corresponding to divacancies whose axes are in [111] direction are longer than the ones located in [100] direction independently of the sublattice they belong to. This result can be understood realizing that [111] direction divacancy is formed by two nearest-neighbours atoms. However, the [100] direction divacancy is formed by two next nearest-neighbours, consequently loosing relative localization of the positron in this divacancy with respect to the divacancy whose axes are in [111] direction.

The lifetime values obtained in both types of divacancies are longer than 190 ps (in the case of divacancies located in [111] direction they are longer than 205 ps) indicating

TABLEIII. – Theoretical divacancy positron lifetime t2 v.

Composition in at.% Divacancy Divacancy’s axes t2 v(ps)

18.75 Zn-12.5 Al VCu-VCu [111] 208.4 18.75 Zn-12.5 Al VCu-VZn [111] 209.5 18.75 Zn-12.5 Al VCu-VAl [111] 208.2 18.75 Zn-12.5 Al VCu-VCu [100] 192.9 18.75 Zn-12.5 Al VZn-VZn [100] 196.1 18.75 Zn-12.5 Al VCu-VAl [100] 194.8 12.5 Zn-18.75 Al VCu-VCu [111] 210.2 12.5 Zn-18.75 Al VCu-VZn [111] 211.5 12.5 Zn-18.75 Al VCu-VAl [111] 210.1 12.5 Zn-18.75 Al VCu-VCu [100] 194.4 12.5 Zn-18.75 Al VZn-VZn [100] 198.3 12.5 Zn-18.75 Al VCu-VAl [100] 197.0

that depending on the type of divacancy, their lifetimes are 10–25 ps longer than the lifetimes found for monovacancies in the alloys with the same composition. Therefore, the relatively close values calculated for positron lifetimes at mono- and divacancies indicate that in case of existence of mono- and divacancies their trapping signal will together contribute to the second component of the lifetime spectra, in agreement with the observed behaviour. These results also indicate that, even though mono- and divacancies are expected to form in quenching experiments from high temperature on b Cu-Zn-Al alloys [7, 8], it is not possible to get such information from the spectra decomposition because, in these experiments, the estimated amount of divacancies formed during quenching is small compared to the quenched-in monovacancies.

6. – Conclusions

The positron bulk lifetime in the ternary alloy b Cu-Zn-Al varies very little with composition. The theoretical calculations predict the largest change is due to the amount of Al content in the alloy.

No preferential positron trapping is expected in the monovacancies of one of the two sublattices. The positron lifetime value depends mainly on and increases with the Al content in the near-neighbourhood of the vacancy. The dependence on the type of vacancy or alloys’ composition is much weaker. The monovacancy positron lifetime predicted by the calculations is close to 180 ps in good agreement with the experimental results.

Two types of divacancies with different positron lifetimes are expected in these alloys. The theoretical calculations predict lifetime values close to 195 ps for the divacancies whose axes are in the [100] direction and 10–15 ps larger for the divacancies whose axes are in the [111] direction.

The calculated lifetimes of positrons trapped at divacancies, relatively close to the lifetimes at monovacancies, suggest the difficulty of discriminating among them from a free decomposition of the experimental positron lifetime spectra. Therefore, in case of existence of mono- and divacancies their trapping signal will together contribute to the second component of the lifetime spectra.

* * *

One of us (FP) would like to thank the Basque Government and DGICYT (Project PB93-1255) for financial support.

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