Analysis of electron-positron momentum spectra of metallic alloys as supported by first-principles calculations

Electron-positron momentum distributions measured by the coincidence Doppler broadening method can be used in the chemical analysis of the annihilation environment, typically a vacancy-impurity complex in a solid. In the present work, we study possibilities for a quantitative analysis, i.e., for distinguishing the average numbers of different atomic species around the defect. First-principles electronic structure calculations self-consistently determining electron and positron densities and ion positions are performed for vacancy-solute complexes in $\mathrm{Al}\text{−}\mathrm{Cu}$, $\mathrm{Al}\text{−}\mathrm{Mg}\text{−}\mathrm{Cu}$, and $\mathrm{Al}\text{−}\mathrm{Mg}\text{−}\mathrm{Cu}\text{−}\mathrm{Ag}$ alloys. The ensuing simulated coincidence Doppler broadening spectra are compared with measured ones for defect identification. A linear fitting procedure, which uses the spectra for positrons trapped at vacancies in pure constituent metals as components, has previously been employed to find the relative percentages of different atomic species around the vacancy [A. Somoza et al. Phys. Rev. B 65, 094107 (2002)]. We test the reliability of the procedure by the help of first-principles results for vacancy-solute complexes and vacancies in constituent metals.

Figure 1

(Color online) Calculated CDB RD spectra for vacancies in elemental strained metals. (a) The solid (red) line shows the result for the vacancy in strained Al and the dashed (green) line for the vacancy in strained Mg. (b) The solid (dark blue) line is the result for a vacancy in strained Cu and the dotted (light blue) line for the vacancy in strained Ag.

Figure 2

(Color online) Experimental CDB spectrum $(\circ )$ (RD to bulk Al) of (a) an as-quenched $\mathrm{Al}\text{−}(1.1\mathrm{at}\%)\mathrm{Cu}$ sample, and (b) the same sample aged $1\min$ at $150°\mathrm{C}$. The dash-dotted (blue), solid (red), and dotted (black) lines show the best one-component fits obtained for vacancy-Cu defects with two, three, and four Cu atoms, respectively.

Figure 3

(Color online) Influence of different approximations on the first-principles results for Al vacancies decorated with Cu atoms. In all panels, the solid (red) line shows the RD spectrum obtained for the Al vacancy surrounded by three Cu atoms on the [100] plane in the strained Al lattice including ionic relaxation due to the electronic structure and the localized positron state. (a) The dash-dotted (dark blue) and the dotted (blue) lines show the results corresponding to ideal (no relaxation) ion positions and for two and three Cu atoms, respectively. (b) The dash-dotted (dark blue) and the dotted (blue) lines correspond to the Al lattice without strain and to three and four Cu atoms, respectively. (c) The dash-dotted (dark blue) curve shows the result for three Cu atoms nearest neighbors to each other. (d) The dash-dotted (dark blue) curve corresponds to the fit using the linear combination of RD spectra for two and four Cu atoms with equal weights.

Figure 4

(Color online) CDB RD spectra for $\mathrm{vacancy}\text{−}\mathrm{Cu}$ complexes in $\mathrm{Al}\text{−}\mathrm{Cu}$ alloys with (a) 2, (b) 4, (c) 8, and (d) 12 nearest neighbor Cu atoms. The tetragonal configuration is used in the case of four Cu atoms. The solid (red) curves give the results of the first-principles calculations. The dashed (green) curves are the linear combinations [Eq. (2)] of the RD spectra calculated for vacancies in strained Al and Cu. The dash-dotted and dotted curves show the Al and Cu contributions, respectively. The inset in (a) shows the magnified low-momentum region.

Figure 5

(Color online) (a) CDB RD spectra for $\mathrm{vacancy}\text{−}\mathrm{Cu}\text{−}\mathrm{Mg}$ complexes in $\mathrm{Al}\text{−}\mathrm{Cu}\text{−}\mathrm{Mg}$ alloys with two Cu and zero, one, and two Mg atoms denoted by solid (red), dotted (dark blue), and dashed (green) lines, respectively. (b) CDB RD spectrum for a vacancy surrounded by two Cu, two Mg, and two Ag atoms and the best linear-combination fit denoted by a solid (red) line and (blue) stars, respectively. The components of the fit, corresponding to the vacancies in strained Al, Cu, Mg, and Ag, are shown by dash-double-dotted (blue) line, dashed (dark blue) line, dash-dotted (pink) line, and long-dotted (green) lines, respectively.

Figure 6

Results of CDB measurements (markers) for (a) $\mathrm{Al}\text{−}(1.1\mathrm{at}\%)\mathrm{Cu}$ alloy measured after quenching and (b) after $1\min$ aging at $150°\mathrm{C}$. The solid lines show the empirical fitting curves.