Determination of Local Order in a Material. EXAFS and SEELFS

     These two techniques exploit the absorption of energy by an atom under the impact of a beam of X-ray photons or high-energy electrons.   The first of these, also the oldest, goes by the name of extended X-ray absorption fine structure (EXAFS). It refers to fine structure spectroscopy in the vicinity of an X-ray absorption threshold. An X-ray photon is absorbed by a given atom in the material, exciting an electron from an inner electron shell to an unoccupied state above the Fermi level, which corresponds to a well-defined energy for each type of atom, whence the chemical sensitivity of the method. The excited atom relaxes by emitting an electron whose wave function interacts with neighbouring atoms. If the atoms are in a crystal lattice, interference will occur between the wave function of the photoelectron and the wave functions of neighbouring atoms. This will cause a change in the absorption of X rays by the target atom (and neighbouring atoms), which will be detected in the form of low amplitude oscillations in the X-ray absorption spectrum near the chosen atomic absorption threshold. A full analysis of this technique can be found in [1].

In practice, carrying out a Fourier transform of these oscillations, one obtains the radial distribution of atoms in the vicinity of the target atom (up to a phase factor). For example, if we consider a solid with NaCl-type structure and examine a sodium threshold, we find a first peak in the radial distribution corresponding to the Na–Cl separation, followed by a second peak corresponding to the Na–Na separation, and a third peak corresponding to the second Na–Cl separation. The intensities of the various peaks are proportional to the number of atoms in the considered coordination sphere. This technique can thus be used to measure the local order in monatomic or multiatomic materials. In contrast to X-ray diffraction,  it can be used on objects with no long-range order or on small clusters. In the latter case, it is extremely useful for determining the lattice parameter (see Fig. 1.).  The technique known as surface extended electron energy loss fine structure  (SEELFS) is analogous to EXAFS, except that the atom is excited by an electron of well-defined energy (usually in the range 3–10 keV) and the energy loss spectrum is measured near an ionisation threshold of the relevant target atom. Since the electrons do not penetrate very far into the material, this technique can only be used to study surfaces or thin films. Spectra are analysed in an analogous way to those produced by EXAFS, but for a quantitative analysis, one must take into account the fact that the excitation is obtained by electrons. The reader is referred to [2] for more details.

Fig. 1 Contraction of the lattice parameter of copper clusters as a function of the reciprocal of their diameter. Circles correspond to measurements of electron energy loss near an ionisation threshold (SEELFS). Taken from De Crescenzi et al. [3]. The straight line shows measurements of X-ray absorption (EXAFS). Taken from Apai et al. [4]

       As can be seen from Fig. 1. the contraction varies linearly with the reciprocal of the particle size. For a diameter of 2 nm, it is 2%. According to (1.1), the gradient of the straight line yields the value of the surface stress as 3.35 J/m2.  The pressure exerted on the crystal lattice is then 6.7GPa, which is extremely high. It is reasonable to ask how far (1.1) remains valid.

        (1.1)

Put another way, can one still appeal to quantities like the specific surface energy and the surface stress, quantities defined in the context of macroscopic thermodynamics, when dealing with nanoscale systems? To address these questions, one may turn to numerical simulation. Indeed, good (semi-empirical) interatomic potentials are available for describing metals, i.e., n-body potentials in which each bond depends on the local atomic environment, in contrast to the so-called pairwise potentials [5–7].

    Figure .2. shows the change in the lattice parameter as a function of the reciprocal of the radius of spherical particles, obtained by numerical simulation using EAM-type (embedded atom method) semi-empirical potentials [8]. The relationship is linear down to a size of about 4a0, where a0 is the lattice parameter. This corresponds to a diameter of 2.5–3 nm. One might expect to find that at smaller sizes the relationship expressed by (1.1) would no longer be valid. However, it seems that this discrepancy is rather due to the fact that the specific surface energy and surface stress are no longer constant. Indeed,  with the same kind of simulation, these two quantities have been calculated for different (spherical) particle sizes and the results do indeed show that they are no longer constant below a diameter of about 2–3 nm. In fact, they increase as the size continues to decrease. One might think that these deviations are due to the constraint, imposed in the calculation, of a spherical particle shape which, as we shall see below, does not correspond to the equilibrium shape of the crystal particles. 

Fig. 2. Change in lattice parameter, relative to the bulk solid, as a function of the reciprocal of the radius for spherical clusters (with fcc structure) of Ag (stars), Au (triangles), Cu (diamonds), and Pt (circles). Numerical simulations at 0 K. Taken from Swaminarayan et al.

References

1. B.K. Teo:EXAFS: Basic Principles, Data Analysis(Springer, Berlin Heidelberg New York 1986)

2. M. De Crescenzi: Surf. Sci. Rep.21, 89 (1995)

3. M. De Crescenzi, M. Diociaiuti, L. Lozzi, P. Picozzi, S. Santucci: Phys. Rev.  B35, 5997 (1987)

4. G. Apai, J.F. Hamilton, J. St¨ohr, A. Thomson: Phys. Rev. Lett.43, 165 (1979)

5. S.M. Foiles, M.I. Baskes, M.D. Daw: Phys. Rev. B33, 7983 (1986)
6. V. Rosato, M. Guillop´e, B. Legrand: Phil. Mag. A59, 321 (1989)
7. K.W. Jacobsen, J.K. Nørskov, M.J. Puska: Phys. Rev. B35, 7423 (1987).

8. S. Swaminarayan, R. Najafabadi, D.J. Solowitz: Surf. Sci.306, 367 (1994).