Electron atomic scattering factors and scattering potentials of crystals

Transcription

1 PERGAMON Micron 30 (1999) Review Electron atomic scattering factors and scattering potentials of crystals L.-M. Peng* Beijing Laboratory of Electron Microscopy, Institute of Physics and Center for Condensed Matter Physics, Chinese Academy of Sciences, P.O. Box 2724, Beijing , People s Republic of China Received 20 February 1999; received in revised form 22 March 1999; accepted 6 April 1999 Abstract The concepts of complex electron atomic scattering factors and principles for evaluating these factors are discussed and their applicability is examined. Numerical procedures and routines for calculating these factors are described, and for 98 neutron atoms and 109 ions the real part of the electron atomic scattering factors were parameterized using 10 and eight parameters, respectively. Procedures for constructing two and three dimensional scattering potentials using the complex atomic scattering factors are illustrated with examples; effects of thermal vibrations of the crystal lattice are discussed Elsevier Science Ltd. All rights reserved. Keywords: Crystal potential; Atomic scattering factor; Debye Waller factors Contents 1. Introduction Basic equations and quantities One-body wave equation for the incident electrons Scattering potential Elastic electron atomic scattering factors Atomic scattering factors of neutral atoms Atomic scattering factors of ions Complex electron atomic scattering factors Basic principles of electron absorption Thermal diffuse scattering and Debye Waller factors Evaluation of TDS absorptive atomic scattering factors Accuracy of the numerical procedure illustrated with an example Constructions of the scattering potentials from atomic scattering factors Scattering potential of a single neutral atom Scattering potential of an assemble of neutral atoms Constructions of scattering potentials of ionic crystals Scattering potential of a single ion Scattering potential of an assemble of ions Effects of charge redistribution in a crystal Summary and discussion Acknowledgements References Introduction * Fax: address: lmpeng@lmplab.blem.ac.cn (L.-M. Peng) The concept of electron atomic scattering factor was introduced and developed in the process of evaluating scattered beam amplitudes of electrons by atoms and crystals. The term atomic scattering factor is indeed misleading in /99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S (99)

2 626 L.-M. Peng / Micron 30 (1999) the sense that this terminology often leads people to relate the atomic scattering factor to the scattered electron beam amplitude from a single atom while it is a well-known fact that only under the unrealistic weak phase object approximation (WPOA) are the two things approximately equal (Cowley, 1990). In this article, we will first discuss the definition and evaluation of electron atomic scattering factors, and show that the usefulness of the concept of electron atomic scattering factor is indeed not limited to the validity of the WPOA or kinematic theory of electron scattering. A useful definition of the electron atomic scattering factor, denoted by f e s, is to regard the factor as the Fourier transform (F) of the electrostatic Coulomb potential of the atom, f(r), i.e. f e s ˆKF{f r }; where K is a constant and in SI units K ˆ 2pm 0 e=h 2 ; m 0 and e are the mass and charge of the electron, and h is the usual Planck constant. For an isotropic atom the atomic electrostatic Coulomb potential is spherically symmetric, i.e. f(r) ˆ f(r), the electron atomic scattering factor is therefore isotropic giving f e s ˆf e s. A closely related quantity to the electron atomic scattering factor is the electron scattering amplitude f(k 0, k) bya single atom. This quantity may be obtained from the asymptotic form of the solution of the problem of an electron scattering by a potential f(r) " # 2 2m 72 f r c r ˆEc r ; 2 in which c(r) is the wave function of an electron, and all other quantities have their usual meaning. Assuming that the incident electron wave function takes the form of a plane wave, after the colliding process when the electron is well separated from the potential f(r), it can be shown (Schiff, 1968) that the asymptotic form of the wave function can be written in a form c r!exp ik r f k 0 ; k exp ikr : 3 r Here the plane wave is an incident along the direction of k, and the electron scattering amplitude f(k 0, k) corresponds to scattering from k to k 0. In the limiting case of a weak potential, i.e. f(r)! 0, the electron scattering amplitude by a single atom is equal to the electron atomic factor, i.e. f e s ˆf k 0 ; k with s ˆ k 0 k =4p. In general these two quantities are different, with f(k 0, k) being a complex entity and f e s being a real function. The only general statement which may be made of the relation between the two quantities is that, given the atomic potential distribution f(r) (which may be obtained via inverse Fourier transform of the electron atomic scattering factor of the atom), the scattering amplitude by the atomic potential f(r) may be obtained via solving the wave equation (2). 1 For an assemble of atoms, the total electrostatic Coulomb potential V(r) may be approximately represented to be a superposition of atomic potentials V r ˆX f i r r i ; 4 i and this approximation is called the superposition approximation, which neglects any effect resulting from a charge redistribution in bringing together isolating atoms in forming a crystal. In the vicinity of nuclei this superposition approximation is an excellent approximation while in covalent crystals the occupation of the bonding states localized in between the nuclei introduces observable features in the diffracted beam amplitudes. In reciprocal space the superposition approximation works well for applications involving large angles of scattering. For small angles of scattering the effect of atom interactions or charge re-distribution is not negligible and may be measured accurately via the X-ray diffraction (Coppens, 1997) and convergent beam electron diffraction (CBED) (Spence and Zuo, 1992). High energy electrons interact with the nuclei and electrons of solids both elastically and inelastically. For elastic scattering involving no energy losses the effective scattering potential seen by the incident high energy electrons is related to the electrostatic Coulomb potential and information concerning both the nuclei coordinates and electron distribution may be obtained. The definitions of the scattering potential and its relation to the atomic scattering factors will be discussed in Section 2, numerical procedures for calculating the electron atomic scattering factors and their representation will be given in Section 3 for neutral atoms and ions. In realistic solids both the nuclei and electrons are not static. The incident electrons will experience a time dependent scattering potential and consequently this time dependence will result in energy losses of the incident electrons. The inelastically scattered electrons will then be lost from the field of elastic scattering. For some applications the effect of the inelastic scattering or energy losses on the elastic scattering may be taken into account by the introduction of a complex scattering potential. The principles of absorption occurring in high energy electron diffraction will be discussed in Section 4, and the validity for the use of the complex scattering potential will be examined. In Sections 5 and 6 the procedures for constructing scattering potentials for various geometries will be given. In Section 7, a brief discussion will be given for the effects of charge redistribution in crystals, and in Section 8, we summarize the concepts and approximate procedures introduced in this article. 2. Basic equations and quantities The problem of electron diffraction by a solid is in general a many-body problem involving both the incident

3 L.-M. Peng / Micron 30 (1999) electrons and the many nuclei and electrons of the solid (Peng, 1995). Of the various interactions, the so-called exchange and correlation interactions are most difficult to deal with and are still the main obstacles for a deeper understanding of such interesting phenomenon as high T c superconductivity (Poole et al., 1995). Fortunately for high energy electrons many of the effects of exchange and correlation may be avoided and the general many-body wave equation involving both the incident electron and the nuclei and electrons of the solid may be reduced to an effective one-body equation for the incident electron. In this section we will discuss the approximations relevant to high energy electron diffraction and the relation of the effective onebody electron scattering potential to that of the atomic potential One-body wave equation for the incident electrons The electron diffracted beam amplitudes may be obtained via solving the wave equation for a many electron system (Parr and Yang, 1989) " # 2 2m 72 H s V Coul V xc C ˆ E t C; 5 in which H s is the Hamiltonian of the solid, V Coul is the electrostatic Coulomb potential describing the interactions between the incident electron and the electrons and the positively charged nuclei of the solid, V xc is the exchangecorrelation potential resulting from electron electron interactions, E t is the total energy and C is the corresponding wave function of the many-electron system, all other physical constants have their usual meaning. In setting up Eq. (5), the evaluation of the exchangecorrelation potential V xc is perhaps the most difficult part. We first consider what this potential means in electron diffraction. The wave function C of the many electron system must be antisymmetric under exchange of any two electrons because the electrons are fermions. The antisymmetry of the wave function produces a spatial separation between electrons that have the same spin and thus reduces the Coulomb energy of the electronic system. The reduction in the energy of the electronic system due to the antisymmetry of the wave function is called the exchange energy. The Coulomb energy of the many electron system may be reduced even further if electrons that have opposite spins are also spatially separated. This reduction in the Coulomb energy must be balanced, however, by an increase in the kinetic energy of the electrons resulting from the deformation of the electronic wave functions in order to separate the electrons. The difference between the many-body energy of the many electrons system and the energy of the system satisfying only the condition of antisymmetry of the wave function is called the correlation energy. For high energy (from a few KeV to a few MeV) electron diffraction, the incident electrons can be effectively distinguished from the electrons of the solid (with energy in the order of less than 100 ev). The exchange-correlation effects, which are important when the incident electron and crystal electrons approach indistinguishability as in the case of a low-energy electron diffraction (LEED), may be neglected (Rez, 1976). For high energy electron diffraction, we may therefore write the total wave function as a product of the two functions, c describing the incident electron and f describing the electrons and nuclei of the solid, i.e. C ˆ cq. Substitution of this relation into Eq. (5), we then obtain the wave equation for the incident electron " # 2 2m 72 V r c r ˆEc r ; 6 here V(r) may be expressed for a system composed of atoms siting at R n Z e 2 r r 0 V r ˆ r r 0 dr0 X n Ze 2 r R n ; where r r 0 ˆPn f n r 0 2 is the total number density of electrons at the point r 0, with f n r being the nth occupied electronic state of the system (Dederichs, 1972; Humphreys, 1979) Scattering potential For the incident high energy electrons, V(r) may be conveniently regarded as the scattering potential. In principle this potential depends on both the positions of the nuclei and the distribution of the electrons around the nuclei via R n and r(r) in Eq. (7), and it is this function which may in principle be retrieved from the elastic scattering experiments. In the limiting case of well separated atoms, we may neglect the overlap of the electron density between neighboring atoms to write r r ˆPn r n r R n and therefore V r ˆX f n r R n ; 8 with n f n r R n ˆ Z e 2 r n r 0 R n r r 0 R n dr 0 Ze2 r R n ; i.e. the total scattering potential may be expressed to be a superposition of contributions from individual atoms. For elastic scattering involving no energy losses, the scattering processes are determined entirely by the Coulomb potential. For a general case involving energy losses, a complex scattering potential shall be used, and a detailed discussion on the complex scattering potential will be given in Section Elastic electron atomic scattering factors For an isolated atom, both the electron density r(r) and the atomic potential f(r) are spherical symmetric, i.e. r(r) ˆ r(r) and f(r) ˆ f(r). The electron atomic scattering factor is 7

4 628 L.-M. Peng / Micron 30 (1999) Table 1 Numerical and fitted electron atomic scattering factors f e s s ˆ 0.01 s ˆ 0.02 s ˆ 0.03 s ˆ 0.04 s ˆ 0.05 s ˆ 0.06 s ˆ 0.07 s ˆ 0.08 s ˆ 0.09 s ˆ 1.0 Numerical(O) Fitted(O) Numerical(Mg) Fitted(Mg) Numer 1 ) Fitted( O1 ) Numerical(Mg2 ) Fitted(Mg 2 ) defined in Eq. (1) as proportional to the Fourier transform of the atomic Coulomb potential f(r), i.e. f e s ˆK Z f r exp iq r dr ˆ 8p2 m 0 e h 2 Z 0 r 2 f r sin 4psr dr; 4psr in which q ˆ 4ps, s ˆ sin u=l, u being the half of the scattering angle and l is the electron wavelength. For high energy electron diffraction the scattering is basically forwarded, i.e. the electron scattering factor has appreciable amplitude only within a small angular range, typically a few degree corresponding to s 2.0 Å 1. The atomic scattering factor for X-ray is defined to be the Fourier transform of the electron density Z f X s ˆ r r exp iq r dr ˆ 4p Z 0 r 2 r r sin 4psr dr: 4psr 9 10 For an atom with an atomic number Z and an electron density distribution r(r), the atomic electrostatic potential is given by f r ˆ 1 ( ) Ze Z er r 0 4pe 0 r r r 0 dr0 ; in which the first term on the right-hand side results from the positive charged nucleus and the second term describes the contribution from electrons associated with the atom. In terms of f (X) (s) and noticing the relation R drexp iq r =r ˆ 4p=q 2 ; we have f e s ˆ 2pm Z 0e h 2 f r exp 4pis r dr ˆ m0e ( 2 Z exp iq r 2h 2 Z dr e 0 r Z ) Z exp iq r r r r 0 e iq r dr 0 0 Š r r 0 d r r 0 ˆ m0e 2 8pe 0 h 2 Z f X s Š s 2 ; and this is the famous Mott formula (Mott and Massey, 1965) relating the atomic scattering factors of electron to that of the X-ray. In the Mott formula if s is measured in Å 1 the above equation then gives f e s ˆ m0e 2 8pe 0 h 2 Z f X s Š s 2 ˆ 0: Z f X s Š s 2 ; 11 and f (e) (s) is given in Å. Both the electron density r(r) and atomic potential f(r) may be calculated using for example the Thomas Fermi Dirac models of the atom or the Hartree Fock Slater program (Herman and Skillman, 1963), the Dirac Slater program and the relativistic-hartree Fock program (Cowley, 1992). Of these the most sophisticated and probably most accurate results are those of Doyle and Turner, 1968 and Rez et al., Listed in Table 1 are the values of f (e) (s) for neutral oxygen and nickel atoms and Ni 2 and O 1 ions for some selected values of s. In the table, the electron atomic scattering factors for neutral atoms were taken from Table 2 Fitting parameters for neutral atoms and ions of magnesium and oxygen a 1 a 2 a 3 a 4 a 5 b 1 b 2 b 3 b 4 b 5 O Mg O Mg

5 L.-M. Peng / Micron 30 (1999) Fig. 1. Numerical and fitted electron atomic scattering factors for neutral atoms of magnesium and oxygen. Table of Cowley, 1992, and that for ions were taken from Table of the same reference Atomic scattering factors of neutral atoms We will first consider the electron atomic scattering factors for neutron atoms. While the numerical values of the scattering factors can be incorporated directly into computer programs for dynamical electron diffraction calculations, there exist many situations where an analytical expression for the scattering factors is desired, as in the case of reflection high energy electron diffraction (RHEED) (Dudarev et al., 1995) and the development of tensor methods for the direct inversion of crystal structures using high energy electrons (Peng, 1997b) and channeling theory (Dyck and de Beeck, 1996). The most widely used analytical approximation to f (e) (s) nowadays is to fit the scattering factors as a sum of n Gaussians f e s ˆXn a i exp b i s 2 ; 12 iˆ1 where a i and b i are the fitting parameters. This approximation was first introduced by Vand et al., 1957 for the X-ray scattering factor, and later by Doyle and Turner, 1968 for the real part of the electron scattering factors. For neutral atoms parameterization has been done using 10 fitting parameters, i.e. n ˆ 5, and the fitting parameters a i and b i have been tabulated for all the elements of the periodic table (Peng et al., 1996c). Listed in Table 1 are some the typical numerical (taken from the International Tables of Crystallography (Cowley, 1992)) and fitted values of f (e) (s) using the parameters given in Table 1 of Peng et al., 1996c for Mg and O atoms and Table 1 of Peng, 1998 for Mg 2 and O 1 ions (for the convenience of our discussions the relevant parameters have been listed in Table 2). Table 1 shows clearly that the analytical fittings for both neutral atoms and ions are excellent, and the final differences between the numerical and fitted values are well within the errors intrinsic to the original numerical procedures for evaluating these factors. Fig. 1 shows some plots of numerical and fitted f (e) (s) for a finer grid of and wider range s. The analytical fitting is seen to be excellent for s up to 2.0 Å 1. Although parameters resulting from 10-parameter fitting were given for all the elements in the periodic table, we noted that many of the crystallography routines originated from the X-ray diffraction community use eight fitting parameters for representing the scattering factors f(s). It is for the benefit of those users of eight fitting parameters, we list in Table 3 the results for the eight-parameter fitting for s up to 2.0 Å 1. In the last column of the table an indicator for the fitting error s is given. This indicator is defined to be the root of the mean square value of the deviation s 2 between the numerical and fitted scattering factors using the parameters given in the table, v 2 3 s ˆ 1 X M f e s M i X4 2 u t 4 a j exp b j s 2 i 5 ; 13 iˆ1 jˆ1 in which M is the total number of the available values of the numerical scattering factors (taken from the International Tables for Crystallography (Cowley, 1992)). The values of s show that the eight-parameter fitting is rather accurate for all the elements, although when comparing to that given in Table 2 of Peng et al., 1996c it is clear that the results

6 630 L.-M. Peng / Micron 30 (1999) Table 3 Eight-parameter fitting of the elastic atomic scattering factors Element Z a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 s H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm

7 L.-M. Peng / Micron 30 (1999) Table 3 (continued) Element Z a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 s H He Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf using 10-parameter fitting (given in Table 1 of Peng et al., 1996c) are in general more accurate than those obtained using the present Table Atomic scattering factors of ions The electron scattering factor of an ion is very different from that of a neutral atom, and in this section we will pay particular attention to the handling of these factors. For X- ray diffraction, the atomic scattering factors of both neutral atoms and ions satisfy the condition that (James, 1965) Z 0 ˆ lim f X s ; 14 s!0 where Z 0 is the number of electrons associated with each atom which can either be in a neutral or in a charged ionic state. For a neutral atom Z 0 ˆ Z, Z being the atomic number of the atom. For an ion Z 0 Z, and the difference between the two quantities represents the excess or deficiency of charge on the nucleus resulting from a charge transfer associated with the formation of chemical bonds in the crystal (Cowley, 1992; Spence and Zuo, 1992). The atomic scattering factor for electron diffraction is related to that for the X-ray diffraction by the Mott formula (11). For an ion where the number of electrons associated with the ion is not equal to the charge of the nucleus Z Z 0, it follows from Eq. (11) that as s approaches zero the scattering factor diverges as Z Z 0 =s 2. Shown in Fig. 2 are the plots of the electron atomic scattering factors for Mg 2 and O 1 ions for s 0.04 Å 1. For smaller s values the electron atomic scattering factors diverge to infinity rapidly. It has been known for some years, see for example Doyle and Turner, 1968 that the divergence of the electron scattering factor of an ion arises from the contribution of unscreened long-range Coulomb potential of the ionic charge on the nucleus. This may be readily demonstrated

8 632 L.-M. Peng / Micron 30 (1999) Fig. 2. Electron atomic scattering factors for Mg 2 and O 1 ions. by rearranging Eq. (11) as f e s ˆ m0e 2 Z 0 f X s 8pe 0 h 2 s 2 m 0e 2 DZ 8pe 0 h 2 s 2 ˆ f e 0 s m 0e 2 8pe 0 h 2 DZ s 2 ; 15 Mg 2 and O 1 ions. The parameters for the fitting were taken from Peng et al., 1998, in which parameters for 109 ions over the entire periodic table have been given, together with an error analysis of the fittings. 4. Complex electron atomic scattering factors where DZ ˆ Z Z 0 represents the ionic charge and the second term on the right-hand side of the above equation represents the divergent contribution from the unscreened Coulomb potential of the ionic charge. The first term on the right-hand side (i.e. f e 0 s ) results from the screened atomic field. The condition (14) ensures that f e 0 s remains finite in the limit s! 0. The electron scattering factors of ions have been calculated numerically and tabulated by several authors including Doyle and Turner, 1968 and Rez et al., In principle ab-initio numerical values obtained by the above authors may be fitted using the same analytical form (12) as for neutral atoms. However, since expression (12) does not diverge for the zero angle of scattering as it should do for an ion, it is not suitable for an accurate fitting of the ionic scattering factors (for a discussion of the errors introduced by this procedure, see Peng et al., Instead the electron scattering factor of an ion is represented as follows: f e s ˆXn jˆ1 a j exp b j s 2 m 0e 2 8pe 0 h 2 DZ s 2 ; 16 in which the finite scattering amplitude of an electron by the screened atomic field f e 0 s is represented as a sum of Gaussians. Shown in Fig. 3 are the numerical and fitted f e 0 s for 4.1. Basic principles of electron absorption For high energy electron diffraction, the incident electrons may be scattered elastically or inelastically by the solid (Cowley, 1993). In an elastic collision the incident electron does not lose any energy and the solid is left in its original state, i.e. f f ˆ f i (here the subscripts f and i denote the final and the initial state, respectively). In contrast, in an inelastical collision the incident electron losses an amount of energy equals DE ˆ E f E i, and the solid is excited from the initial state f i to a final state f f (Egerton, 1986; Wang, 1995). To a good approximation, the effect of an inelastic scattering on the elastic scattering may be taken into account by regarding the interaction between the incident electron and the solid as a complex potential. This complex potential is usually called the optical potential, in analogy with the long-standing use of a complex refractive index for discussing the optical properties of partially absorbing media (Yoshioka, 1957). After an inelastic collision the incident electron has lost energy while the solid is excited to a higher energy state. For high energy electrons, the probability that the inelastically scattered electron reappears in the elastic channel is very small (Rez, 1976). As far as the elastic scattering is concerned, the inelastically scattered electron

9 L.-M. Peng / Micron 30 (1999) Fig. 3. Numerical and fitted electron atomic scattering factors for the finite part f e 0 and for Mg 2 and O 1 ions. has been absorbed, and the inelastic scattering events contribute only an imaginary addition to the optical potential. In principle the inelastically scattered electrons may be separated from the elastically scattered electrons using an energy filter, and the propagation of the elastically scattered electrons through the crystal may be described using a complex scattering potential. For most of the energy filters installed in transmission electron microscopes (TEM) the energy resolution (about ev) is much larger than the typical energy losses caused by phonon excitations. The elastic electrons and those suffered from phonon excitations are therefore indistinguishable in the final image plane. In diffraction contrast (DC) imaging of crystal defects the inelastically scattered electrons are separated from those of the elastically scattered electrons by the use of a very small aperture around the transmitted (bright field) and one of the Fig. 4. Schematic diagram showing the principles of electron absorption occurring in the modes of (a) DC imaging of crystal defects and (b) HREM imaging of crystal lattices.

10 634 L.-M. Peng / Micron 30 (1999) Fig. 5. Schematic diagram showing the effects of diffuse background in the CBED. diffracted beam (dark field) (Hall and Hirsch, 1965) (see Fig. 4(a)). Since the phonon excitations usually involve relatively large momentum transfer perpendicular to the incident and diffracted beams, the use of a small aperture is particularly efficient for filtering out those electrons which have suffered energy losses caused by the phonon excitations. Energy filtered DC images of crystal defects may therefore be well described by the use of the optical potential. In high resolution imaging of crystal lattices, however, a larger aperture is usually used which allows some of the inelastically scattered electrons to contribute to the final high-resolution electron microscopy (HREM) images (see Fig. 4(b)). While the inelastically scattered electrons hardly interfere with those of the elastically scattered electrons, these inelastically scattered electrons and in particular the phonon scattered electrons may interfere with themself and form high resolution features which are superimposed on the usual HREM images formed by the elastically diffracted electrons (Cowley, 1988). For very thin samples and qualitative analysis of the images, the effect of the inelastically scattered electrons which have passed through the objective aperture on the elastically scattered electrons may be neglected and those blocked by the aperture may be represented by the use of the complex potential. For quantitative treatments of the inelastically scattered electrons on the elastically scattered beams, a density matrix approach may be used, see for example Dudarev et al., 1993a. The most accurate quantitative electron diffraction work so far carried our has been done using the CBED mode of an electron diffraction (Spence and Zuo, 1992). In this mode a convergent electron beam, usually defined by a circular aperture, is focused on the specimen (see Fig. 5). Each diffraction spot of the conventional electron diffraction pattern is then spread into a circular disk, and each point in the disk corresponds to a particular angle of incidence. Fig. 5 shows that while for each incidence there is a diffuse background superimposed in the elastic spot diffraction pattern, the total diffuse scattered electron distribution is a smooth function compared with the much more rapidly varying curves of the elastically scattered electron beam intensity and may be approximated as a constant within each diffraction disk and be subtracted from the elastic CBED pattern Thermal diffuse scattering and Debye Waller factors Of all the mechanisms of inelastic scattering for high energy electrons, thermal diffuse scattering (TDS), plasmon and single electron excitations account for much of the effect of the inelastically scattered electrons on the elastic electron wave field. The plasmon scattering results from the long-range Coulomb interaction of the incident electron and the conduction electrons (hardly bound to individual atoms) of the solid. The conduction electrons form the so-called electron gas in the crystal and the density of the electron gas oscillates against a background of fixed positive ions, and the quanta of the oscillations are called plasmon. The important characteristics of the plasmon scattering is that the scattering potential due to the plasma excitations is essentially uniform over the crystal. In the electron diffraction experiments this means that the plasma excitations contribute only to the mean absorption, and in most of the transmission electron diffraction (TED) experiments this

11 L.-M. Peng / Micron 30 (1999) contribution does not affect the intensity distribution of the diffracted beams and may be neglected. Single electron excitations result from the short-range interaction between the incident electron and the more localized electrons around the nuclei. As discussed above the long-range interaction between the incident electron and the electrons of the solid may excite plasma, i.e. collective motions of the conduction electrons, and these motions screen out the long-range Coulomb interaction in the solid. The remaining part of the interaction is therefore of a short-range in nature, typically extend over one to two interatomic distances. An important process of inelastic single electron scattering involves the excitation of an inner shell electron associated with an ion core. Since the scattering potential associated with the excitations of these tightly bound core electrons are localized, large momentum transfers are expected and in principle this type of processes may affect the intensity distribution of the diffracted beams. The total cross-section of the single electron excitations involving large momentum transfer is, however, rather small compared to that of the TDS. In real crystals the positions of the nuclei are not fixed at their lattice sites, but subject to oscillations about their equilibrium positions. To a good approximation the probability distribution of an atom around a lattice site may be described by a so-called probability density function P(r) such that the time average of the atomic potential f(r) around the lattice point is given by spreading out the atom with a spread function (James, 1965; Cowley, 1990) 1 P r ˆ 2pa 2 3=2 exp r2 =2a 2 ; where a is the root mean-square deviation of the atom from its lattice site. In reciprocal space this effect is equivalent to multiply the atomic scattering function by the Fourier transform of the spread function such that for the averaged atomic potential we have f e s!exp 4p 2 a 2 s 2 f e s : The prefactor exp 4p 2 a 2 s 2 in the above equation is the Debye Waller factor. In many texts a usual B-factor B ˆ 4p 2 a 2 is introduced such that the Debye Waller factor is rewritten as exp Bs 2, and in what follows we shall refer to the B-factor as the Debye Waller B-factor. To a good approximation the effect of thermal motion of the lattice is therefore accounted for by the Debye Waller factor. Effectively for an atom in a real crystal the effective atomic scattering factor equals that of an isolating atom multiplied by a temperature factor, and the later temperature factor is defined as T ˆ exp Bs 2 : 17 For a simple cubic lattice the Debye Waller B-factor is isotropic and may be represented by a simple number. In a general case the Debye Waller factor is a matrix, and we will discuss this point in more detail in the next section. The importance of the effect of thermal motion in crystallography cannot be overemphasized. A Temperature Factor project was indeed initiated in 1985 by the Neutron Diffraction Commission of the International Union of Crystallography [Acta Cryst. (1985) B41, 374], and compilations have been made of Debye Waller factors of several cubic elements (Butt et al., 1988), cubic compounds (Butt et al., 1993) and hexagonal close packed elements (Gopi and Sirdeshmukh, 1998). The recommended values of B are given, however, only at a few fixed temperatures, e.g. at 293 K, and these values are usually difficult to be interpolated into other temperatures at which real electron diffraction experiments are conducted. Alternatively the Debye Waller factors may be calculated for all temperatures using one of the models of lattice dynamics. For elemental crystals with only one atom per unit cell, the temperature dependent Debye Waller factors may be obtained given the phonon density of states g(v) (Lovesey, 1984) B ˆ 4p2 m Zv m 0 coth v 2k B T g v dv; v 18 in which m is the mass of the atom, T is temperature, k B is the usual Boltzman constant, and v m is the maximum phonon frequency. Experimentally the phonon density of states g(v) may be measured by the inelastic neutron scattering (Bruuesch, 1987). A direct numerical integration of the above equation then gives the Debye Waller factor for any temperature. Indeed this has been done for 68 elemental crystals over the temperature range from 1 to 1000 K or the melting temperature of the crystal whichever is smaller (Peng et al., 1996b; Gao and Peng, 1999). For majority of compounds, however, only the phonon dispersion curves (i.e. the phonon frequencies v as functions of the momentum transfer q) are known. An example of the phonon dispersion curves is shown in Fig. 6 for a NiO single crystal. The lattice dynamics of ionic crystals, such as NiO, may be modeled excellently by the so-called shell mode (Dick and Overhauser, 1958; Cochran, 1959; Bilz and Kress, 1979). By adjusting several force constants the phonon frequencies resulting from the shell model may be fitted into the experimentally measured phonon dispersion curves. Both the frequencies v q;j and the displacement vector U(q, j) can be obtained for all branches (j ˆ 1,,3n, n being the number of atoms within each unit cell) and all phonon wave vectors q. For cubic lattice, the Debye Waller factors are isotropic. The Debye Waller factor for each of the atoms in the unit cell is given by B ˆ 8p2 X E qj 3mN qj v 2 U q; j 2 ; 19 qj

12 636 L.-M. Peng / Micron 30 (1999) zinc-blende-structure materials and Debye Waller B- factors were obtained over the temperature range from 1 to 1000 K (Reid, 1983; Gao and Peng, 1999). Recently the lattice dynamics of 19 compounds with the sodium chloride structure was investigated and the Debye Waller factors at any temperatures were obtained (Gao et al., 1999). An example of the Debye Waller factor as a function of temperature is given in Fig. 7 for a single crystal of CaO, together with the experimental values at room temperature. In the figure the Debye Waller factor was calculated using a shell model (Gao et al., 1999) and that of the experimental values were taken from Butt et al., This figure shows that excellent agreement has been achieved between the experimental and calculated Debye Waller factors Evaluation of TDS absorptive atomic scattering factors Fig. 6. Experimental and calculated phonon dispersion curves along the [111] direction for a NiO crystal. In the figure LO and TO refer to the longitudinal and transverse optical branches of thermal vibrations, respectively, and LA and TA to that of the acoustic branches. where E qj ˆ v qj n qj 1 2 is the mean energy of the phonon in the state (qj), and n qj is the mean occupation number of phonons in the state (qj) and is given by the Bose Einstein distribution n qj ˆ 1 exp v qj =k B T 1 : The shell model has been used for obtaining phonon frequencies and the displacement vectors for 17 We now turn to the evaluation of the complex electron scattering factors f (e) (s) f e s ˆRe{f e s } i Im{f e s }: The real part of the scattering factor Re{f (e) (s)} is the usual scattering factor as discussed in Section 3, but note that here the TDS effect has been taken into account via the Debye Waller factor. The imaginary part is the absorptive scattering factor resulting from the inelastic scattering processes. As has been discussed in the previous section and shown by many authors that for all g 0, the atomic contribution to the absorptive scattering factor is dominated by the TDS (Whelan, 1965b; Humphreys and Hirsch, 1968; Radi, 1970), and the absorptive scattering factor may therefore be approximated as the TDS scattering amplitude. Numerically the TDS contribution can be calculated readily following the original formulation of Hall and Hirsch, 1965 assuming isotropic Debye Waller factors and an Einstein model for the TDS scattering (Bird and King, 1990; Fig. 7. Debye Waller B-factor as a function of temperature for CaO.

13 L.-M. Peng / Micron 30 (1999) Weickenmeier and Kohl, 1991; Rossouw and Bursill, 1985) Im{f e s } f e TDS s T s ˆ 2h Z ds 0 f j s m 0 v 2 s0 j f j s 2 s0 j T s T s 20 s s0 T 2 i s0 ; 2 where v is the velocity of the electron, T(s) is the temperature factor given by Eq. (17), and the Debye Waller B- factor is related to the mean-square thermal vibration amplitude u 2 via B ˆ 8p 2 u 2 : 21 It should be noted that although the expression (20) was originally derived based on an Einstein model of the TDS (Hall and Hirsch, 1965), the applicability of this expression is not indeed limited to this model. This is because, while the vibrations of atoms in a crystal are strongly correlated (Born and Huang, 1954) and the correlation affects the distribution of the diffusely scattered electrons (Dudarev et al., 1991; Dudarev et al., 1993b), the total effect of correlation on the absorptive electron scattering factor can be shown to be negligible (Dudarev et al., 1992; Peng, 1997a). For the general case of anisotropic thermal vibrations, we have an identical form (Peng, 1997a) for the TDS absorptive scattering factor, but with the temperature factor T(s) now given by T s ˆexp{ G T bg}: 22 Let g ˆ 4ps ˆ hb 1 kb 2 lb 3, G is then a 3 1 column vector and its transpose is given by G T ˆ h; k; l : The matrix b is a symmetric matrix 0 1 u 2 1 u 1 u 2 u 1 u 3 b ˆ 1 u 2 2 u 1 u 2 B 2 u 2 u 3 A ; u 3 u 1 u 3 u 2 u and is usually referred to as the mean-square displacement matrix. In the X-ray crystallography the general anisotropic vibration parameters are usually given as the elements of a U matrix which are related to that of the b matrix by the following relation b ij ˆ 2p 2 U ij b i b j : 24 Explicitly the anisotropic temperature factor is given by T s ˆexp{ 2p 2 U 11 hb 1 2 U 22 kb 2 2 U 33 lb 3 2 2U 12 hb 1 kb 2 2U 13 hb 1 lb 3 2U 23 kb 2 lb 3 }: 25 Experimentally U ij may be obtained by fitting quantitatively the calculated X-ray beam intensities with the experimentally measured X-ray intensities using the general anisotropic temperature factor (Giacovazzo, 1992). The absorptive scattering factor (20) depends in general on the acceleration voltage E via v occurring in Eq. (20), the Debye Waller B-factor, and the angle of scattering via s. The dependence on the acceleration voltage can be factored out and the results can be converted easily from one voltage (for example 100 kev) to any other voltage via f e b 100 kev TDS s; Ev ˆ f e TDS s; 100 kev ; b Ev 26 p in which b ˆ 1 g 2 ; g ˆ m=m 0 ˆ1:0 1: E, and the acceleration voltage E is measured in kev. The dependence of the absorptive scattering factor on the angle of scattering or s can also be fitted using five Gaussians as we did for the elastic scattering factor in Section 3 f e TDS s exp Bs2 =2 ˆX5 iˆ1 a TDS i exp b TDS i s 2 : 27 The inclusion of the factor exp( Bs 2 /2) on the left-hand side of Eq. (27) is due to the fact that the TDS absorptive scattering factor (20) increases with s as exp(bs 2 /2) at large angles of scattering, while the fitted quantity f e TDS s exp Bs2 =2 vanishes with increasing s. In general for neutral atoms the complex electron atomic scattering factors are given by f e s ˆXn iˆ1 {a i exp b i s 2 ia TDS i B=2 s 2 Š} exp Bs 2 : exp b TDS i 28 For ions an additional term resulting from ionic charges needs to be added to the above expression. Shown in Fig. 8 are the calculated TDS electron absorptive atomic scattering factors for the neutral atoms of Mg and O, their analytical fitting using 10 parameters and results obtained using the so-called proportional model. The computations were done for 100 kev primary beam energy, using the room temperature Debye Waller factors B ˆ Å 2 for Mg and B ˆ Å 2 for oxygen (Zuo et al., 1997). This figure shows firstly that the 10 parameters s curves for both Mg and O are perfect, and this is also true for other elements for typical primary beam energies at a range of temperatures. Secondly, the proportional model of the TDS as originally proposed for Al by Hashimoto et al., 1962 does not seem to work for the case of MgO at all. This is because the shape of the analytical fitting of the f e TDS s curve is much broader than that of the corresponding real part of the electron atomic scattering factor, indicating that the TDS absorptive potential in real space is much more localized than the real Coulomb potential. In the figure the real parts of f (e) (s) for Mg and O atoms have been scaled to equal that of the TDS parts at zero angle of scattering, and the scaling constants are very different for Mg (184.34) and O (133.74) and are more than 10 times larger than that f e TDS

14 638 L.-M. Peng / Micron 30 (1999) Fig. 8. Numerical and analytical fitting of the TDS absorptive electron atomic scattering factors for neutral Mg and O atoms, and normalized real part of the electron atomic scattering factors. originally proposed for Al. Since both the numerical values and analytical fitting parameters are nowadays readily available, we suggest that the proportional model of the absorptive potential should not been used for any quantitative work. Shown in Fig. 9 are the real and imaginary parts of the averaged potential of an Ag single crystal, with the average being taken over the plane parallel to the (001) surface. The calculations were made for a 20 kev primary beam energy and the crystal was assumed to be at the room temperature. The imaginary part of the potential is many times smaller than the real part, and in the figure this part is magnified four times such that it is shown in the figure with the same order of magnitude as the real part. The figure shows clearly that the imaginary part of the potential is much more localized to the unclei than the real part, and this is consistent with our earlier conclusion based on the electron atomic scattering factors. A major difficulty encountered in the analytical parameterization of the imaginary part or absorptive scattering Fig. 9. Real and imaginary parts of the averaged potential V 0 (z) calculated for Ag(001) surface and 20 kev primary beam energy. The room temperature Debye Waller factor was used.

15 L.-M. Peng / Micron 30 (1999) Fig. 10. Energy filtered experimental Si[110] zone axis CBED pattern. The pattern was obtained for a primary beam energy of kev, an energy winder of 10 ev and an electron probe size of 1.4 nm, using a Philips CM200/FEG electron microscope. factors f e TDS s using such conventional algorithms as Marquardt Levenberg procedure (Press et al., 1986) is that the results so obtained depend sensitively on the initially assigned values of the fitting parameters. This situation may not seem to be so serious for the fitting of the elastic scattering factors, as the number of data sets are limited. However, for the absorptive scattering factors a more robust procedure is needed. This is because for a given element the absorptive scattering factors depend both on the angles of scattering and the Debye Waller factors. It is difficult to make an exhaustive tabulation of all the parameters for all elements and compounds. A robust computer program has been developed by Peng et al., 1996c) which is able to return automatically the required fitting parameters given the name of the element, the acceleration voltage and the Debye Waller factor. Parameterization for absorptive atomic scattering factors and 100 kev primary beam energy has been made for 17 important materials with the zinc-blende-structure and 44 elemental crystals over a wide range of temperatures for which the accurate Debye Waller factors are available (Peng et al., 1996b, 1996c). The effects of the crystal ionicity on the absorptive atomic scattering factors have been investigated numerically by Peng, It was found that these factors differ from that of the corresponding neutral atoms noticeably only for small angles of scattering with s 0.3 Å 1. For most applications of the TED the effect of crystal ionicity on the absorptive crystal structure factors may be neglected and the corresponding factors of the neutral atoms may be used [Peng, 1998] Accuracy of the numerical procedure illustrated with an example Before concluding this section we will first demonstrate the accuracy of the numerical procedure for evaluating the complex electron atomic scattering factor with an example. Shown in Fig. 10 is an experimental [110] zone axis energy filtered CBED pattern taken from a silicon single crystal, using a nominal primary beam energy of 200 kev. This pattern was obtained using a Philips CM200/FEG electron microscope and a small probe of 1.4 nm. As the illuminated crystal area is very small, it is then reasonable to assume that the CBED pattern resulted from an area of uniform thickness and orientation, free of crystal defects, and is therefore well suited for comparison by calculations. The experimental pattern of Fig. 10 was recorded using a slow-scan CCD camera of the size of There exist in a single pattern more than one million independent experimental data points. As the number of structural parameters of a single crystal of silicon is much less than the number of the available experimental data points, we choose to extract only a subset of the available data for comparison with calculations. In the present case the subset contains data lying on two intersecting lines pointing along the 1 11Š and [002] directions, and the intensity variation curves or rocking curves are shown in Fig. 11 together with the theoretically calculated rocking curves. The goodnessof-fit between the experimental and theoretical intensities is described by a merit function x 2 ˆ X k 1 s 2 k I exp k I cal k Š 2 ; 29 in which s 2 k denotes the variance of the kth experimental data point, I exp k and I cal k refer to experimental and calculated diffracted beam intensities, respectively. The variance of the experimental data s 2 k may be estimated using the experimentally measured values of detector quantum efficiency for different beam intensities, see

16 640 L.-M. Peng / Micron 30 (1999) be expressed analytically as f e s ˆXn iˆ1 a i exp b i s 2 exp Bs 2 : 31 Substitution of Eq. (31) into Eq. (30) then gives f r ˆ ˆ ˆ h2 2pm 0 e 1 X n 2p 3 iˆ1 h2 X n Z 2pa 2pm 0 e i iˆ1 0 exp b i B 4p 2 q2 dq h2 X n 2pm 0 e i a i Z a i exp b i B s 2 Š exp iq r dq q 2 Zp 0 exp iqr cos u d cos u 4p 3=2 exp 4p2 r 2 b i B b i B! ; 32 Fig. 11. Energy-filtered experimental and fitted Si[110] CBED rocking curves for (a) a line scan along the [111] direction and (b) a line scan along the [200] direction of Fig.10. The boundaries in (b) correspond to AB of Fig. 10 for the (000) disk and CD for the (111) disk, respectively. for example Ren et al., In general the x 2 function depends on a set of crystal structure factors, i.e. {F g }, and these structure factors may be obtained by adjusting these parameters until the best fit is achieved between the theoretical and experimental data. For the experimental data shown in Fig. 11 a minimum x 2 value of 1.4 was obtained. As an idea experiment free from any systematic errors would give an idea x 2 value of 1.0, our results suggest that the procedure for evaluating the absorptive structure factors, especially for those associated with larger momentum transfer, is rather accurate and that systematic errors have been minimized. 5. Constructions of the scattering potentials from atomic scattering factors 5.1. Scattering potential of a single neutral atom For a single isolated atom, its electrostatic potential may be obtained readily via inverse Fourier transform of the atomic scattering factor f r ˆ h2 2pm 0 e 1 2p 3 Z f e s exp iq r dq; 30 with q ˆ 4ps. As discussed in the previous section that the atomic scattering factor (including the effect of TDS) may where the prefactor h 2 /2pm 0 e ˆ if f(r) is measured in electron volts. This equation shows clearly that the atomic potential is spherically symmetric and may be expressed as a sum of Gaussians. Shown in Fig. 12 are two atomic potential plots as a function of distance from the center of the atom. The plots were calculated using Eq. (32) for a static atom with B ˆ 0, and for a vibrating atom with an averaged deviation distance u ˆ 0.1 Å. This figure shows that the Debye Waller factor or thermal vibration tends to spread the potential. This is because the width of each Gaussian of Eq. (32) increases from b i for a static atom to b i B for a vibrating atom, and this real space observation is consistent with the loss of the high frequency components of the atomic scattering factors in reciprocal space due to thermal vibrations Scattering potential of an assemble of neutral atoms In general the difference between the real crystal potential and that of a model crystal composed of isolating atoms dv(r) is very small and varies slowly within the crystal. For most of the inorganic crystals and many applications it is a good starting point to use the superposition approximation and write the crystal potential as X V r ˆX f i r R n r i ; 33 n i in which the index n refers to the nth unit cell of the crystal, and i denotes ith atom in the unit cell (see Fig. 13). For a three-dimensional (3D) periodic crystal, we can expand the potential in terms of a set of plane waves associated with the 3D reciprocal lattice vectors {g} V r ˆX V g exp ig r : 34 g

17 L.-M. Peng / Micron 30 (1999) Fig. 12. Time averaged atomic potential plots for a static (solid) and vibrating (dotted) He atom. The gth Fourier coefficient of the potential is given by V g ˆ 1 Z V r exp ig r dr; 35 V where V is the volume of the crystal. Substituting Eq. (33) into Eq. (35) and noting the relation that exp ig R n ˆ1, we obtain V g ˆ 1 ( X X Z f V i r R n r i n i ) exp ig r R n r i Š dr exp ig r i ˆ N V ( ) X Z f r 0 exp ig r 0 dr 0 exp ig r i ; i 36 where N is the total number of unit cells in the crystal. By substituting Eq. (30) into Eq. (36) and writing V/N ˆ V c, here V c is the volume of a unit cell, we then obtain for a 3D periodic crystal potential h 2 V g ˆ F 2pm 0 V g ; 37 c with F g ˆ X i f e i s exp ig r i 38 being the usual crystal structure factor for gth reflection. For the RHEED, it is convenient to use a mixed real and reciprocal space representation of the crystal potential V x; z ˆX V G z exp ig x ; 39 G where {G} are two-dimensional (2D) reciprocal lattice vectors parallel to the surface. The 2D Fourier coefficients V G (z) are given by V G z ˆ 1 S Z V x; z exp ig x dx; 40 in which S is the illuminated surface area by the incident electron beam. When dealing with 2D periodic crystals, it is useful to consider the crystal as consisting of a stack of atomic layers with index l (Kambe, 1967; Zhao et al., 1988). In analogous to Eq. (33), we can write the crystal potential as follows: X X V r ˆX f i r x n r i d l ; 41 n i l where n are the indices of the 2D unit cells, r i ˆ (x i, z i ) and i denotes the ith atom in the unit cell, d l ˆ x l ; z l denotes the separation of the origin of the lth atomic layer from the origin of the 0th layer (see Fig. 14). Substitution of Eq. (41) into Eq. (40), noticing that exp ig x n ˆ1, then gives V G z ˆ 1 X Z f S i x 0 ; z 0 exp ig x 0 dx 0 l;n;i exp ig x i x l Š; 42 where z 0 ˆ z z i z l. We now consider the integral in the big bracket. Substituting expression Eq. (30) for f i (r) and Eq. (31) for f e i s

18 642 L.-M. Peng / Micron 30 (1999) Fig. 13. Index system for a 3D crystal. In the figure n are the indices of the unit cells, and i are that of the atoms within one unit cell. into Eq. (42), we obtain Z Z { }dx 0 ˆ f x 0 ; z 0 exp ig x 0 dx 0 ( h 2 Z Z X n ˆ 2p 4 a j exp b! j B g 2 m 0 jˆ1 4p 2 ) exp ig r 0 dg exp ig x 0 dx 0 ˆ h 2 2p 4 m 0 Z X n jˆ1 a j exp b! j B q 2 g 2 z 4p 2 Z exp ig z z 0 dg exp i q G x 0 Š dx 0 ˆ 4p 2 m 0 X n a j jˆ1 s p b j B " # exp 4p2 z 02 : b j B " exp b j B G 2 4p 2 # 43 By writing N as the total number of surface unit cells within the illuminated area, S 0 ˆ S/N as the area of a single surface unit cell, and substituting Eq. (43) into Eq. (42), we then obtain V G z ˆ 4p S 0 2 m 0! X i;l X n jˆ1 a i j s b i j p B i " exp b i j B i G 2 # 4p 2 " # exp 4p2 b i j B z z i i z l 2 exp ig x i x l Š: 44 For G ˆ 0; V G z ˆV 0 z, representing the averaged potential distribution of a crystal along the surface normal (z) direction. The plots shown in Fig. 9 are examples of this type of potential distributions for the real and imaginary parts of the potential, respectively. 6. Constructions of scattering potentials of ionic crystals 6.1. Scattering potential of a single ion For a single ion with ionic charge DZ, the electron atomic scattering factor is given by Eq. (16) f e s ˆX5 jˆ1 a j exp b j s 2 m 0e 2 8pe 0 h 2 DZ s 2 : The 3D inverse Fourier transform of this scattering factor then gives the real space representation of the scattering potential due to a single ion f r ˆ DZe2 4pe 0 r f 0 r ; 45 in which the first term on the right-hand side of the equation represents the ionic contribution to the potential, and the second term denotes that of the remaining atomic field and this contribution is given by expression (32). In general the ionic contribution represents a long-range interaction, while the field f 0 (r) is only short-ranged representing contribution from a screened atomic field. Shown in Fig. 15 are plots of the total atomic potentials of Ni 2 and O 2 ions (solid curves), and that of the screened potential, i.e. f 0 (r) (dotted curves). For the negative ions it is seen that the potential field may be positive around the ions siting at the origin. In general the atomic potential of an ion is more spread compared

19 L.-M. Peng / Micron 30 (1999) Fig. 14. Index system for a 2D crystal. In the figure n are the indices of the surface unit cells, i are that of the atoms within one unit cell, and l are that of the 2D layers parallel to the surface. with that of a neutral atom, and for an isolated ion the potential diverges at the center of the ion. In real crystals, however, the condition of the charge neutrality requires that equal number of positive and negative point charges must be present in the crystal, and this condition ensures that the potential is finite everywhere inside and outside the crystals Scattering potential of an assemble of ions For an idea assembly of isolating ions, the total scattering potential may again be written as a sum of contributions from individual ions. For a 3D periodic crystal the procedure for constructing the potential is the same as for as an assembly of neutral atoms. In practice all specimens used in the electron diffraction are not 3D and may better be regarded to be 2D, either in the form of a slab as used in the TEM electron microscopy or effectively a semi-infinite bulk crystals as used in the RHEED geometry of electron diffraction. A simple analytical expression for V G (z) as Eq. (44) is not available for ionic crystals due to the difficulty in dealing with the un-screened long-range Coulomb potential of the ionic charge. On a purely numerical basis the scattering Fig. 15. Plots of the total atomic potentials for Ni 2 and O 2 ions (solid line) and the corresponding screened atomic potentials (dotted) associated with these ions.

20 644 L.-M. Peng / Micron 30 (1999) Fig. 16. Averaged potential distributions for (a) a crystal composed of neutral Ni and O atoms and (b) an ionic crystal composed of Ni 2 and O 2 ions. potential of an ionic crystal may be constructed using a 3D super unit cell (Peng et al., 1996a; Peng et al., 1998). The size of the super unit cell in the plane parallel to the surface equals the size of the surface unit cell, and it may be arbitrarily large in the direction normal to the surface. Using the super unit cell the potential can be calculated using conventional Fourier expansion both in the plane of the surface and in the direction normal to it. Let g ˆ (G, l), where G are the 2D reciprocal lattice vectors and l are the components of the 3D reciprocal lattice vectors normal to the surface. After all the components V g have been calculated, V G (z) can then be obtained V G z ˆX V G;l exp ilz : 46 l It is convenient to separate V G z into two parts, namely the contribution due to ionic charge DV G z and the remaining part associated with the screened atomic field VG z : 0 While the contribution resulting s always remains finite, the part of the potential associated with the presence of ionic charge may give rise to divergent terms. For the termination of the crystal lattice satisfying the condition of charge neutrality and having zero total dipole moment, the averaged potential is given by Peng et al., 1998 from f e 0 DV 0 z ˆ X l 0 DV 0 l exp ilz e2 X DZ 4pV i 8p 2 z 2 i B i : i 47 Three cases may be distinguished for the surfaces of ionic crystals. In the first case, both positive and negative ions are located in the same plane and this leads to the cancellation of a long-range Coulomb contribution to the crystal potential. The second class of ionic surfaces consists of layers of ions having net non-zero charge. Ionic layers may be grouped into repeat units of planes of the form (anion) (cation) (anion) characteristic of the (111) surface of the fluorite structure, so that each repeat unit satisfies the charge neutrality condition and also has zero net dipole moment perpendicular to the surface. For this class of surfaces the Coulomb term leads to unusual features in the distribution of the potential in the crystal. The third type of the termination of an ionic crystal lattice gives rise to a sequence of repeat units characterized by a non-zero net dipole moment resulting in a divergent behavior of the potential. Shown in Fig. 16 are the two one-dimensional (1D) potential distributions for a NiO crystal. The potentials are averaged parallel to the (001) surface of the NiO crystal, corresponding to the first case as discussed in the previous paragraph. In this particular case the effect due to the ionic charges cancels each other out. The ionic nature of the crystal, however, is seen to affect the crystal inner potential noticeably. It has been shown (Peng et al., 1998) that charge transfer between lattice sites occurring in an ionic crystal affects very substantially the form of RHEED rocking curves, and particular attention should be paid in investigating the surfaces of ionic crystals (Peng et al., 1997). 7. Effects of charge redistribution in a crystal When isolating atoms are brought together to form a solid, the charge density of the core electrons of the isolating atoms will be deformed and the out valence electrons will be redistributed to form bonds which glues the atoms together to form the crystal. In general the electrostatic Coulomb

21 L.-M. Peng / Micron 30 (1999) Dirac Fock package of Grant et al., 1980 and Rez et al., The resulting electron atomic scattering factor f (e) (s) may be represented either numerically over a fixed grid of the s values (Cowley, 1992) or analytically using some Gaussians. For neutral atoms the atomic potential is shortranged and the analytical fitting is straightforward f e s ˆXn i a i exp b i s 2 : Fig. 17. (110) cross-sections: (a) the total charge density of a silicon single crystal; and (b) the charge deformation map. potential may be written as V r ˆV 0 r dv r ; 48 in which the first term represents contribution from the isolating atoms and the second term describes the deviation of the real potential from the superposition approximation. Experimentally the Fourier coefficients V g of the crystal potential may be measured accurately via the CBED experiments, and these coefficients can be converted into that for X-ray and a charge density map can then be obtained via Fourier transform of these coefficients. For a given crystal structure the corresponding r(r) map is called the total charge density map (see Fig. 17(a)) and the map dr(r) ˆ r(r) r 0 (r) is called the deformation map reflecting how valence electrons are redistributed within the crystal and therefore how atoms are brought together forming the crystal. It should be noted that in general dr(r) is a slow varying function, meaning that it affects usually only the lowest order reflections, but not higher order reflections. For Si the effect vanishes rapidly and becomes smaller than the measurement error for reflections higher than (331). 8. Summary and discussion The electron atomic scattering factor f (e) (s) is defined as proportional to the Fourier transform of the atomic electrostatic potential f(r). For an isolating atom, the atomic potential f(r) is spherically symmetric, and so is the electron atomic scattering potential, i.e. Z f e s ˆK f r exp 4pis r dr; where K is a constant. For an isolating atom, the atomic potential f(r) and therefore electron atomic scattering factors may be calculated accurately using for example the multiconfiguration A complete table containing all the fitting parameters for all the elements in the periodic table has been given by Peng et al., 1996c) for n ˆ 5, and in this article for n ˆ 4. For an ion, the atomic potential may be partitioned into a longrange part due to the ionic charge DZ contribution and a short-range part resulting from the remaining screened atomic potential, and the corresponding atomic scattering factor may be represented analytically as f e s ˆXn i a i exp b i s 2 0: DZ s 2 ; where the first term on the right-hand side represents the finite contribution due to a short-range atomic potential and the second term denotes that of the ionic charge and which is divergent at zero angle of scattering, i.e. in the vicinity of s ˆ 0. A table containing 10 fitting parameters, i.e. a i ; b i, i ˆ 1,,5, for 109 ions has been given by Peng, The numerical values of f (e) (s) can be obtained readily using the above expression, in which s is measured in Å 1 and f (e) (s) is in unit of Å. Thermal vibration of a lattice tends to spread the atomic potential. Effectively this effect of thermal vibration on the atomic scattering potential may be represented by a convolution of the atomic potential by a Gaussian spread function in real space and equivalently in reciprocal space by multiplying the usual electron scattering factor f e 0 s by a temperature factor T, i.e. f e s!f e s T: For a cubic lattice the temperature factor is isotropic and may be represented by the Debye Waller B-factor as T ˆ exp Bs 2 and for a general case the lattice vibration is anisotropic and the temperature factor is a second-order tensor (Willis and Pryor, 1975; Peng, 1997a). For crystals with cubic lattice, compilations have been made of the Debye Waller factors over a wide temperature range for 17 zinc-blende-structure materials (Reid, 1983; Gao and Peng, 1999), 66 elemental crystals (Peng et al., 1996b; Gao and Peng, 1999), and 19 compounds with the sodium chloride structures (Gao et al., 1999). The incident high energy electrons may also be scattered inelastically by for example phonon, plasma and atomic inner shell electronic excitations. To an excellent approximation the effect of inelastically scattered electrons on the