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Attenuation of excited electrons at crystal surfaces
Journal of Electron Spectroscopy and Related Phenomena 180 (2010) 66–68
Contents lists available at ScienceDirect
Journal of Electron Spectroscopy and Related Phenomena journal homepage: www.elsevier.com/locate/elspec
Attenuation of excited electrons at crystal surfaces I. Bartoˇs a,∗ , E.E. Krasovskii b,c,d a
Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnicka 10, 162 53 Prague 6, Czech Republic Departamento de Física de Materiales, Facultad de Ciencias Químicas, Universidad del Pais Vasco, 20080 San Sebastián, Basque Country, Spain c Donostia International Physics Center (DIPC), 20018 San Sebastián, Basque Country, Spain d IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain b
a r t i c l e
i n f o
Article history: Received 12 January 2010 Received in revised form 13 April 2010 Accepted 23 April 2010 Available online 21 May 2010 Keywords: Surface sensitivity of electron spectroscopies Electron attenuation anisotropy Complex band structure Low energy electron diffraction Photoelectron diffraction
a b s t r a c t Attenuation of the electron current determines surface sensitivity of electron spectroscopies. Pronounced energy and directional dependencies of the electron attenuation in crystals differ strongly from those commonly used for amorphous solids. Quantum descriptions of the electron attenuation can be obtained from the complex band structure of crystal surfaces at energies above the vacuum level. Contributions, specific for concrete processes, are obtained from theoretical description of photoelectron spectroscopy and of electron diffraction. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Surface sensitivity of electron spectroscopies and of electron diffraction results from a short mean free path of low energy electrons (with energies ranging from tens to hundreds eV). The energy dependence is usually described by the ‘universal curve’ with a minimum at around 50 eV. For surface studies the electron attenuation length (AL) represents an important quantitative measure of the surface sensitivity. When an exponential decay of the electron current in the elastic channel is assumed, I(z) = exp(−z/), it can be characterized by the coefficient : AL is then a distance from the surface (depth) on which the electron current propagating from or to the surface gets reduced by a factor of e. Experimental determinations of the AL, mostly by the overlayer method [1], are not easy and reliable. Theoretical simulations based on semi-classical Monte Carlo simulations are suitable for disordered systems only, as diffraction effects are not included. Diffraction effects, which result from coherent quantum interferences, have to be taken into account when dealing with crystals. In crystals, an anisotropy of the electron attenuation and its nontrivial energy dependence are to be expected. Quantummechanical description of the electron propagation at energies above the vacuum level, which incorporate multiple electron scat-
∗ Corresponding author. Tel.: +420 220318599; fax: +420 283343184. E-mail address: [email protected] (I. Bartoˇs). 0368-2048/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2010.04.008
terings with atoms, provide appropriate schemes taking diffraction effects into account. Electron attenuation analysis, performed here on the Cu(1 1 1) surface shows the role of individual branches of the complex band structure (partial waves decomposition) above the vacuum level. Specifics of the investigated physical processes have to be taken into account, namely the role of different initial conditions in electron attenuation: localized electrons excited from a core level in photoemission and delocalized reflected electron wave in low energy electron diffraction. The initial conditions will be shown to affect the angular anisotropy of the electron attenuation. The main features resulting from electron propagation to the crystal surface remain impressed in the resulting angular distributions of electron beams in vacuum. 2. Bloch function attenuation in a semi-infinite crystal The electron attenuation is due to inelastic and elastic collisions of propagating electrons [1]. Inelastic collisions resulting mainly from the electron–electron and electron–phonon interactions are responsible for finite lifetimes of electrons in their excited states. Their effect can be incorporated into the standard theoretical oneelectron approximation schemes by adding imaginary component into the effective crystal potential (optical potential). Elastic collisions, classically speaking, are responsible for the prolongation of the total path traveled to the surface because a zigzag trajectory is longer than a straight line. Quantum mechanically, these interactions are described by the crystal potential and correspond-
I. Bartoˇs, E.E. Krasovskii / Journal of Electron Spectroscopy and Related Phenomena 180 (2010) 66–68
67
situations, of course. This implies that the band structure alone does not suffice to determine the electron attenuation in a specific physical measurement: the appropriate initial conditions have to be specified in addition and we will bring corresponding solutions for LEED and angular resolved photoemission in the next two sections. Several conclusions can be drawn from Fig. 1:
Fig. 1. Energy dependence of the penetration depth for the partial waves (branches of the complex band structure) that most strongly contribute to the LEED state at the normal incidence on Cu(1 1 1). Vertical thickness is proportional to the current carried by the individual partial wave.
ing electron band structure. When dealing with surfaces, not only Bloch waves with real k vectors, but also those with nonzero imaginary component perpendicular to the surface have to be taken into account (complex band structure). These electron states decay then exponentially to the bulk of the crystal. The self-consistent potential with the inverse k·p method in the extended LAPW formalism has been adopted (with the optical potential Vi = 2 eV) for the evaluation of the complex band structure of the Cu(1 1 1) surface, see Ref. [2] for details. Resulting electron band structure of a combined effect of inelastic and elastic electron scatterings Fig. 1 shows electron damping in the individual branches of the complex band structure for k|| = 0 for electron energies 5–160 eV above the Fermi level. Branches at the top belong to waves with longest spatial decay (smallest imaginary component of kperp ) and are responsible for the overall electron attenuation at a given energy. Coupling of individual branches of the complex band structure to the normally incident electron plane wave representing the physical conditions of the low energy electron diffraction (LEED) is indicated by the thickness of the vertical shading of the curves in Fig. 1. This shows weights of individual branches in this experiment; the weights will be different in other experimental
– non-exponential decay of the total wave function in general (only approximately exponential with just one dominant contribution to the total wave function). – oscillating energy dependence of electron attenuation, which is in a strong contrast to a smooth energy dependence of the ‘universal curve’. – no pronounced enhancement of the electron attenuation length with increasing energy. 3. Electron attenuation from LEED theory In LEED, a dynamical theory of electron diffraction is needed for a realistic evaluation of intensities of electron beams diffracted from crystal surfaces: this theory takes into account multiple electron collisions of electrons with atoms in the surface region of a crystal. Calculation of intensities of electron beams back diffracted from crystal surfaces starts from electron scatterings by individual atoms, continues by scatterings by atomic layers and finally by a stack of these layers representing a semi-infinite crystal. The components of this procedure provide description of the electron transport in surface region as well and they can be used to extract the energy dependence and angular anisotropy of the attenuation length [3]. The decrease of the electron beam intensity for the normally incident electron plane wave has been evaluated for two high-symmetry azimuths of the Cu(1 1 1) surface. The calculated spatial decay can be fitted by the exponential providing thus attenuation length magnitudes shown in Fig. 2a and compared with the results of simulations for amorphous copper. The energy dependence of the AL between 30 and 350 eV has oscillating character reflecting the electron band structure in contrast to a smooth energy dependence for amorphous material. Main features of (E) from Fig. 2a, in particular the minimum at 90 eV followed by one broad and two smaller peaks, are rather similar to those observed in the Bloch-waves calculation of the energy depen-
Fig. 2. (a) Energy dependence of the electron attenuation length from LEED for Cu(1 1 1) at = 0◦ and 35◦ compared with a smooth energy dependence of the amorphous Cu. (b) Polar anisotropy of the electron attenuation length of 320 eV electrons for two azimuths compared with a smooth dependence of the amorphous Cu.
68
I. Bartoˇs, E.E. Krasovskii / Journal of Electron Spectroscopy and Related Phenomena 180 (2010) 66–68
dence of the electron attenuation in Fig. 1: the minimum is due to the main current-carrying branch changing from ˛ to , and the two narrower maxima are due to the branch ı. When repeating the procedure for off-normal electron incidence angles the polar angular anisotropy of is obtained. The results, at electron energy 320 eV, show a rather strong anisotropy which is qualitatively different for the two azimuths (Fig. 2b). These anisotropies will be interpreted in the next section devoted to the angular resolved core level photoemission. 4. Electron attenuation from photoelectron diffraction In photoelectron diffraction, the intensities of photoemitted electrons are recorded as a function of energy or exit angle. Photoexcitation from atomic core levels enables a useful chemical sensitivity. A finite atomic cluster is used for the description of the photoemission from the surface layers of crystals. The photoexcitation and subsequent multiple scatterings of electrons by neighboring atoms then reflect local atomic arrangements of the excited atoms in photoemitted electron intensities. The interpretation of local crystallography gets simplified at higher electron energies (usually above around 500 eV) due to electron focusing effects caused by a predominantly forward scattering of electrons by atoms at these energies [4]. Most common experimental investigation of the electron attenuation is realized by the overlayer technique when gradually thicker overlayers of the investigated material are deposited on a substrate and the decrease of electron photoemission from the substrate is recorded. Theoretical determination [5,6] goes similarly analyzing the attenuation of the photoelectron flux excited from gradually deeper and deeper lying atomic layers. From fitting the decreasing intensities by exponentials the angle resolved attenuation lengths can be extracted. The results of the angular resolved electron attenuation of photoemission from the Cu 2p3/2 level from subsurface layers of the Cu(1 1 1) surface for one higher symmetry azimuth are given in Fig. 3a. Relatively smooth angular dependence of photoemission from the topmost atomic layer distinguishes this contribution from those from deeper lying atomic planes. For the azimuth = 0◦ there is an enhanced photoemission intensity at around = 35◦ , which can be attributed to the highly packed row connecting atoms from nearest atomic planes. Fig. 3b shows the polar anisotropy of for two azimuthal angles = 0◦ and 60◦ for photoelectrons with kinetic energy 320 eV. A proper choice of the photoelectron exit direction can thus be utilized to focus the angular resolved photoemission investigation to the desired subsurface plane [7]. The electron ALs for the two high-symmetry azimuths Fig. 3b show a rather strong anisotropy in polar angle dependence, which deviates from the monotonous dependence expected for a homogeneous sample. Angular anisotropies of AL for LEED and photoelectron diffraction are qualitatively similar but differ quantitatively. Their similarity results from forward focusing effects given by the crystal structure whereas the differences reflect different initial conditions for electron propagation to the surface in the two processes: delocalized wave in LEED and localized core level in photoelectron diffraction. 5. Conclusions The electron attenuation in crystals does not simply follow the ‘universal curve’ for the electron mean free path. Diffraction effects are responsible for oscillations of the AL magnitude with electron
Fig. 3. Photoelectron Cu 2p3/2 intensities, excited by Mg K␣ radiation (electron kinetic energy 320 eV): (a) total, surface (L0) and five subsurface layer-resolved contributions (L1–L5) to the polar plots of photoemitted electrons from the Cu(1 1 1) surface for azimuth = 0◦ , (b) polar anisotropy of the effective electron attenuation length for = 0◦ and 60◦ .
energy and for a strong spatial anisotropy as demonstrated here for Cu(1 1 1). Standard XPS quantitative analyses of chemical composition of surfaces may thus provide rather incorrect results when applied to crystalline surfaces. Due to forward focusing effect a proper choice of photoelectron emission direction can be used to get the information predominantly from a desired subsurface atomic plane. Acknowledgments The authors acknowledge the support by the Grant Agency of the Czech Republic (Grant No. 20/07/0601) and by the Institutional Research Plan No. AVOZ10100521. References [1] N. Barrett, E.E. Krasovskii, J.-M. Themlin, V.N. Strocov, Phys. Rev. B 71 (2005) 035427. [2] E.E. Krasovskii, W. Schattke, P. Jiˇríˇcek, M. Vondráˇcek, O.V. Krasovska, V.N. Antonov, A.P. Shpak, I. Bartoˇs, Phys. Rev. B 78 (2008) 165406. [3] O. Romanyuk, I. Bartoˇs, Surf. Sci. 603 (2009) 2789. [4] C.S. Fadley, in: R.Z. Bachrach (Ed.), Synchrotron Radiation Research, Advances in Surface and Interface Science, vol. 1, Plenum, New York, 1992. [5] I. Bartoˇs, Surf. Sci. 603 (2009) 369. [6] F.J. García de Abajo, M.A. Van Hove, C.S. Fadley, Phys. Rev. B 63 (2001) 075404. [7] F. Matsui, T. Matsushita, Y. Kato, M. Hashimoto, K. Inaji, F.Z. Guo, H. Daimon, Phys. Rev. Lett. 100 (2008) 207201.
Contents lists available at ScienceDirect
Journal of Electron Spectroscopy and Related Phenomena journal homepage: www.elsevier.com/locate/elspec
Attenuation of excited electrons at crystal surfaces I. Bartoˇs a,∗ , E.E. Krasovskii b,c,d a
Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnicka 10, 162 53 Prague 6, Czech Republic Departamento de Física de Materiales, Facultad de Ciencias Químicas, Universidad del Pais Vasco, 20080 San Sebastián, Basque Country, Spain c Donostia International Physics Center (DIPC), 20018 San Sebastián, Basque Country, Spain d IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain b
a r t i c l e
i n f o
Article history: Received 12 January 2010 Received in revised form 13 April 2010 Accepted 23 April 2010 Available online 21 May 2010 Keywords: Surface sensitivity of electron spectroscopies Electron attenuation anisotropy Complex band structure Low energy electron diffraction Photoelectron diffraction
a b s t r a c t Attenuation of the electron current determines surface sensitivity of electron spectroscopies. Pronounced energy and directional dependencies of the electron attenuation in crystals differ strongly from those commonly used for amorphous solids. Quantum descriptions of the electron attenuation can be obtained from the complex band structure of crystal surfaces at energies above the vacuum level. Contributions, specific for concrete processes, are obtained from theoretical description of photoelectron spectroscopy and of electron diffraction. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Surface sensitivity of electron spectroscopies and of electron diffraction results from a short mean free path of low energy electrons (with energies ranging from tens to hundreds eV). The energy dependence is usually described by the ‘universal curve’ with a minimum at around 50 eV. For surface studies the electron attenuation length (AL) represents an important quantitative measure of the surface sensitivity. When an exponential decay of the electron current in the elastic channel is assumed, I(z) = exp(−z/), it can be characterized by the coefficient : AL is then a distance from the surface (depth) on which the electron current propagating from or to the surface gets reduced by a factor of e. Experimental determinations of the AL, mostly by the overlayer method [1], are not easy and reliable. Theoretical simulations based on semi-classical Monte Carlo simulations are suitable for disordered systems only, as diffraction effects are not included. Diffraction effects, which result from coherent quantum interferences, have to be taken into account when dealing with crystals. In crystals, an anisotropy of the electron attenuation and its nontrivial energy dependence are to be expected. Quantummechanical description of the electron propagation at energies above the vacuum level, which incorporate multiple electron scat-
∗ Corresponding author. Tel.: +420 220318599; fax: +420 283343184. E-mail address: [email protected] (I. Bartoˇs). 0368-2048/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2010.04.008
terings with atoms, provide appropriate schemes taking diffraction effects into account. Electron attenuation analysis, performed here on the Cu(1 1 1) surface shows the role of individual branches of the complex band structure (partial waves decomposition) above the vacuum level. Specifics of the investigated physical processes have to be taken into account, namely the role of different initial conditions in electron attenuation: localized electrons excited from a core level in photoemission and delocalized reflected electron wave in low energy electron diffraction. The initial conditions will be shown to affect the angular anisotropy of the electron attenuation. The main features resulting from electron propagation to the crystal surface remain impressed in the resulting angular distributions of electron beams in vacuum. 2. Bloch function attenuation in a semi-infinite crystal The electron attenuation is due to inelastic and elastic collisions of propagating electrons [1]. Inelastic collisions resulting mainly from the electron–electron and electron–phonon interactions are responsible for finite lifetimes of electrons in their excited states. Their effect can be incorporated into the standard theoretical oneelectron approximation schemes by adding imaginary component into the effective crystal potential (optical potential). Elastic collisions, classically speaking, are responsible for the prolongation of the total path traveled to the surface because a zigzag trajectory is longer than a straight line. Quantum mechanically, these interactions are described by the crystal potential and correspond-
I. Bartoˇs, E.E. Krasovskii / Journal of Electron Spectroscopy and Related Phenomena 180 (2010) 66–68
67
situations, of course. This implies that the band structure alone does not suffice to determine the electron attenuation in a specific physical measurement: the appropriate initial conditions have to be specified in addition and we will bring corresponding solutions for LEED and angular resolved photoemission in the next two sections. Several conclusions can be drawn from Fig. 1:
Fig. 1. Energy dependence of the penetration depth for the partial waves (branches of the complex band structure) that most strongly contribute to the LEED state at the normal incidence on Cu(1 1 1). Vertical thickness is proportional to the current carried by the individual partial wave.
ing electron band structure. When dealing with surfaces, not only Bloch waves with real k vectors, but also those with nonzero imaginary component perpendicular to the surface have to be taken into account (complex band structure). These electron states decay then exponentially to the bulk of the crystal. The self-consistent potential with the inverse k·p method in the extended LAPW formalism has been adopted (with the optical potential Vi = 2 eV) for the evaluation of the complex band structure of the Cu(1 1 1) surface, see Ref. [2] for details. Resulting electron band structure of a combined effect of inelastic and elastic electron scatterings Fig. 1 shows electron damping in the individual branches of the complex band structure for k|| = 0 for electron energies 5–160 eV above the Fermi level. Branches at the top belong to waves with longest spatial decay (smallest imaginary component of kperp ) and are responsible for the overall electron attenuation at a given energy. Coupling of individual branches of the complex band structure to the normally incident electron plane wave representing the physical conditions of the low energy electron diffraction (LEED) is indicated by the thickness of the vertical shading of the curves in Fig. 1. This shows weights of individual branches in this experiment; the weights will be different in other experimental
– non-exponential decay of the total wave function in general (only approximately exponential with just one dominant contribution to the total wave function). – oscillating energy dependence of electron attenuation, which is in a strong contrast to a smooth energy dependence of the ‘universal curve’. – no pronounced enhancement of the electron attenuation length with increasing energy. 3. Electron attenuation from LEED theory In LEED, a dynamical theory of electron diffraction is needed for a realistic evaluation of intensities of electron beams diffracted from crystal surfaces: this theory takes into account multiple electron collisions of electrons with atoms in the surface region of a crystal. Calculation of intensities of electron beams back diffracted from crystal surfaces starts from electron scatterings by individual atoms, continues by scatterings by atomic layers and finally by a stack of these layers representing a semi-infinite crystal. The components of this procedure provide description of the electron transport in surface region as well and they can be used to extract the energy dependence and angular anisotropy of the attenuation length [3]. The decrease of the electron beam intensity for the normally incident electron plane wave has been evaluated for two high-symmetry azimuths of the Cu(1 1 1) surface. The calculated spatial decay can be fitted by the exponential providing thus attenuation length magnitudes shown in Fig. 2a and compared with the results of simulations for amorphous copper. The energy dependence of the AL between 30 and 350 eV has oscillating character reflecting the electron band structure in contrast to a smooth energy dependence for amorphous material. Main features of (E) from Fig. 2a, in particular the minimum at 90 eV followed by one broad and two smaller peaks, are rather similar to those observed in the Bloch-waves calculation of the energy depen-
Fig. 2. (a) Energy dependence of the electron attenuation length from LEED for Cu(1 1 1) at = 0◦ and 35◦ compared with a smooth energy dependence of the amorphous Cu. (b) Polar anisotropy of the electron attenuation length of 320 eV electrons for two azimuths compared with a smooth dependence of the amorphous Cu.
68
I. Bartoˇs, E.E. Krasovskii / Journal of Electron Spectroscopy and Related Phenomena 180 (2010) 66–68
dence of the electron attenuation in Fig. 1: the minimum is due to the main current-carrying branch changing from ˛ to , and the two narrower maxima are due to the branch ı. When repeating the procedure for off-normal electron incidence angles the polar angular anisotropy of is obtained. The results, at electron energy 320 eV, show a rather strong anisotropy which is qualitatively different for the two azimuths (Fig. 2b). These anisotropies will be interpreted in the next section devoted to the angular resolved core level photoemission. 4. Electron attenuation from photoelectron diffraction In photoelectron diffraction, the intensities of photoemitted electrons are recorded as a function of energy or exit angle. Photoexcitation from atomic core levels enables a useful chemical sensitivity. A finite atomic cluster is used for the description of the photoemission from the surface layers of crystals. The photoexcitation and subsequent multiple scatterings of electrons by neighboring atoms then reflect local atomic arrangements of the excited atoms in photoemitted electron intensities. The interpretation of local crystallography gets simplified at higher electron energies (usually above around 500 eV) due to electron focusing effects caused by a predominantly forward scattering of electrons by atoms at these energies [4]. Most common experimental investigation of the electron attenuation is realized by the overlayer technique when gradually thicker overlayers of the investigated material are deposited on a substrate and the decrease of electron photoemission from the substrate is recorded. Theoretical determination [5,6] goes similarly analyzing the attenuation of the photoelectron flux excited from gradually deeper and deeper lying atomic layers. From fitting the decreasing intensities by exponentials the angle resolved attenuation lengths can be extracted. The results of the angular resolved electron attenuation of photoemission from the Cu 2p3/2 level from subsurface layers of the Cu(1 1 1) surface for one higher symmetry azimuth are given in Fig. 3a. Relatively smooth angular dependence of photoemission from the topmost atomic layer distinguishes this contribution from those from deeper lying atomic planes. For the azimuth = 0◦ there is an enhanced photoemission intensity at around = 35◦ , which can be attributed to the highly packed row connecting atoms from nearest atomic planes. Fig. 3b shows the polar anisotropy of for two azimuthal angles = 0◦ and 60◦ for photoelectrons with kinetic energy 320 eV. A proper choice of the photoelectron exit direction can thus be utilized to focus the angular resolved photoemission investigation to the desired subsurface plane [7]. The electron ALs for the two high-symmetry azimuths Fig. 3b show a rather strong anisotropy in polar angle dependence, which deviates from the monotonous dependence expected for a homogeneous sample. Angular anisotropies of AL for LEED and photoelectron diffraction are qualitatively similar but differ quantitatively. Their similarity results from forward focusing effects given by the crystal structure whereas the differences reflect different initial conditions for electron propagation to the surface in the two processes: delocalized wave in LEED and localized core level in photoelectron diffraction. 5. Conclusions The electron attenuation in crystals does not simply follow the ‘universal curve’ for the electron mean free path. Diffraction effects are responsible for oscillations of the AL magnitude with electron
Fig. 3. Photoelectron Cu 2p3/2 intensities, excited by Mg K␣ radiation (electron kinetic energy 320 eV): (a) total, surface (L0) and five subsurface layer-resolved contributions (L1–L5) to the polar plots of photoemitted electrons from the Cu(1 1 1) surface for azimuth = 0◦ , (b) polar anisotropy of the effective electron attenuation length for = 0◦ and 60◦ .
energy and for a strong spatial anisotropy as demonstrated here for Cu(1 1 1). Standard XPS quantitative analyses of chemical composition of surfaces may thus provide rather incorrect results when applied to crystalline surfaces. Due to forward focusing effect a proper choice of photoelectron emission direction can be used to get the information predominantly from a desired subsurface atomic plane. Acknowledgments The authors acknowledge the support by the Grant Agency of the Czech Republic (Grant No. 20/07/0601) and by the Institutional Research Plan No. AVOZ10100521. References [1] N. Barrett, E.E. Krasovskii, J.-M. Themlin, V.N. Strocov, Phys. Rev. B 71 (2005) 035427. [2] E.E. Krasovskii, W. Schattke, P. Jiˇríˇcek, M. Vondráˇcek, O.V. Krasovska, V.N. Antonov, A.P. Shpak, I. Bartoˇs, Phys. Rev. B 78 (2008) 165406. [3] O. Romanyuk, I. Bartoˇs, Surf. Sci. 603 (2009) 2789. [4] C.S. Fadley, in: R.Z. Bachrach (Ed.), Synchrotron Radiation Research, Advances in Surface and Interface Science, vol. 1, Plenum, New York, 1992. [5] I. Bartoˇs, Surf. Sci. 603 (2009) 369. [6] F.J. García de Abajo, M.A. Van Hove, C.S. Fadley, Phys. Rev. B 63 (2001) 075404. [7] F. Matsui, T. Matsushita, Y. Kato, M. Hashimoto, K. Inaji, F.Z. Guo, H. Daimon, Phys. Rev. Lett. 100 (2008) 207201.