Intensities of diffraction beams

Vainshtein, B. K.; Zvyagin, B. B.

doi:10.1107/97809553602060000558

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 308-309 | 1 | 2 |

Section 2.5.4.3. Intensities of diffraction beams

B. K. Vainshtein^c ^‡ and B. B. Zvyagin^d ^‡

2.5.4.3. Intensities of diffraction beams

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The intensities of scattering by a crystal are determined by the scattering amplitudes of atoms in the crystal, given by (see also Section 5.2.1 ) $[\eqalign{ f_{e}^{\rm abs} (s) &= 4\pi K \int\varphi (r) r^{2} {\sin sr \over sr} \ \hbox{d}r\hbox{;}\cr K &= {2\pi me \over h^{2}}\hbox{; }f_{e} = K^{-1} f_{e}^{\rm abs},\cr} \eqno(2.5.4.13)]$ where $[\varphi (r)]$ is the potential of an atom and $[s = 4\pi (\sin \theta)/\lambda]$ . The absolute values of $[f_{e}^{\rm abs}]$ have the dimensionality of length L. In EDSA it is convenient to use $[f_{e}]$ without K. The dimensionality of $[f_{e}]$ is [potential $[L^{3}]$ ]. With the expression of $[f_{e}]$ in V Å³ the value $[K^{-1}]$ in (2.5.4.13) is 47.87 V Å².

The scattering atomic amplitudes $[f_{e} (s)]$ differ from the respective $[f_{x} (s)]$ X-ray values in the following: while $[f_{x} (0) = Z]$ (electron shell charge), the atomic amplitude at [s = 0] $[f_{e} (0) = 4\pi {\textstyle\int} \varphi (r) r^{2} \ \hbox{d}r \eqno(2.5.4.14)]$ is the `full potential' of the atom. On average, $[f_{e} (0) \simeq Z^{1/3}]$ , but for small atomic numbers Z, owing to the peculiarities in the filling of the electron shells, $[f_{e} (0)]$ exhibits within periods of the periodic table of elements `reverse motion', i.e. they decrease with Z increasing (Vainshtein, 1952, 1964). At large $[(\sin \theta)/\lambda]$ , $[f_{e} \simeq Z]$ . The atomic amplitudes and, consequently, the reflection intensities, are recorded, in practice, up to values of $[ (\sin \theta)/\lambda \simeq 0.8\hbox{--}1.2]$ $[\hbox{\AA}^{-1}]$ , i.e. up to $[ d_{\min} \simeq 0.4\hbox{--} 0.6\;\hbox{\AA}]$ .

The structure amplitude $[\Phi_{hkl}]$ of a crystal is determined by the Fourier integral of the unit-cell potential (see Chapter 1.2 ), $[\Phi_{hkl} = {\textstyle\int\limits_{\Omega}} \varphi ({\bf r}) \exp \{2\pi i ({\bf r} \cdot {\bf h})\} \hbox{ d}v_{r}, \eqno(2.5.4.15)]$ where Ω is the unit-cell volume. The potential of the unit cell can be expressed by the potentials of the atoms of which it is composed: $[\varphi ({\bf r}) = {\textstyle\sum\limits_{{\rm cell}, \, i}} \varphi_{{\rm at}\; i} ({\bf r} - {\bf r}_{i}). \eqno(2.5.4.16)]$ The thermal motion of atoms in a crystal is taken into account by the convolution of the potential of an atom at rest with the probability function $[w({\bf r})]$ describing the thermal motion: $[\varphi_{\rm at} = \varphi_{\rm at} ({\bf r}) * w ({\bf r}). \eqno(2.5.4.17)]$ Accordingly, the atomic temperature factor of the atom in a crystal is $[{f_{eT} [(\sin \theta) / \lambda] = f_{e}f_{T} = f_{e} [(\sin \theta) / \lambda] \exp \{- B [(\sin \theta) / \lambda]^{2}\},} \eqno(2.5.4.18)]$ where the Debye temperature factor is written for the case of isotropic thermal vibrations. Consequently, the structure amplitude is $[\Phi_{hkl} = {\textstyle\sum\limits_{{\rm cell}, \, i}} f_{eT_{i}} \exp \{2\pi i (hx_{i} + ky_{i} + lz_{i})\}. \eqno(2.5.4.19)]$ This general expression is transformed (see IT I, 1952) according to the space group of a given crystal.

To determine the structure amplitudes in EDSA experimentally, one has to use specimens satisfying the kinematic scattering condition, i.e. those consisting of extremely thin crystallites. The limit of the applicability of the kinematic approximation (Blackman, 1939; Vainshtein, 1964) can be estimated from the formula $[A = \lambda \left|{\langle \Phi_{{\bf h}} \rangle \over \Omega}\right| t {{\lower7pt\hbox{$\lt$}}\atop{\raise2pt\hbox{$\sim$}}} 1, \eqno(2.5.4.20)]$ where $[\langle \Phi_{{\bf h}} \rangle]$ is the averaged absolute value of $[\Phi_{{\bf h}}]$ (see also Section 5.2.1 ). Since $[\langle \Phi_{{\bf h}} \rangle]$ are proportional to $[Z^{0.8}]$ , condition (2.5.4.20) is better fulfilled for crystals with light and medium atoms. Condition (2.5.4.20) is usually satisfied for textured and polycrystalline specimens. But for mosaic single crystals as well, the kinematic approximation limit is, in view of their real structure, substantially wider than estimated by (2.5.4.20) for ideal crystals. The fulfillment of the kinematic law for scattering can be, to a greater or lesser extent, estimated by comparing the decrease of experimental intensity $[I_{h} [(\sin \theta)/\lambda]]$ averaged over definite angular intervals, and sums $[{\textstyle\sum}\; f_{\rm obs}^{2} [(\sin \theta)/\lambda]]$ calculated for the same angular intervals.

For mosaic single-crystal films the integral intensity of reflection is $[I_{{\bf h}} = j_{0} S\lambda^{2} \left|{\Phi_{{\bf h}} \over \Omega}\right|^{2} {td_{{\bf h}} \over \alpha} \simeq \Phi_{{\bf h}}^{2} d_{{\bf h}}\hbox{;} \eqno(2.5.4.21)]$ for textures $[I_{{\bf h}} = j_{0} S\lambda^{2} \left|{\Phi_{{\bf h}} \over \Omega}\right|^{2} {tL\lambda p \over 2\pi R' \sin \varphi} \simeq \Phi_{{\bf h}}^{2} p / R'. \eqno(2.5.4.22)]$ Here $[j_{0}]$ is the incident electron-beam density, S is the irradiated specimen area, t is the thickness of the specimen, α is the average angular spread of mosaic blocks, R′ is the horizontal coordinate of the reflection in the diffraction pattern and p is the multiplicity factor. In the case of polycrystalline specimens the local intensity in the maximum of the ring reflection $[I_{{\bf h}} = j_{0} S\lambda^{2} \left|{\Phi_{{\bf h}} \over \Omega}\right|^{2} {td_{{\bf h}}^{2} p \Delta S \over 4\pi L\lambda} \simeq \Phi_{{\bf h}}^{2} d_{{\bf h}}^{2} p \eqno(2.5.4.23)]$ is measured, where ΔS is the measured area of the ring.

The transition from kinematic to dynamic scattering occurs at critical thicknesses of crystals when $[A \geq 1]$ (2.5.4.20). Mosaic or polycrystalline specimens then result in an uneven contribution of various crystallites to the intensity of the reflections. It is possible to introduce corrections to the experimental structure amplitudes of the first strong reflections most influenced by dynamic scattering by applying in simple cases the two-wave approximation (Blackman, 1939) or by taking into account multibeam theories (Fujimoto, 1959; Cowley, 1981; Avilov et al. 1984; see also Chapter 5.2 ).

The application of kinematic scattering formulae to specimens of thin crystals (5–20 nm) or dynamic corrections to thicker specimens (20–50 nm) permits one to obtain reliability factors between the calculated $[\Phi^{\rm calc}]$ and observed $[\Phi^{\rm obs}]$ structure amplitudes of $[ R = 5\hbox{--} 15\%]$ , which is sufficient for structural determinations.

With the use of electron diffractometry techniques, reliability factors as small as [R =] 2–3% have been reached and more detailed data on the distribution of the inner-crystalline potential field have been obtained, characterizing the state and bonds of atoms, including hydrogen (Zhukhlistov et al., 1997, 1998; Avilov et al., 1999).

The applicability of kinematics formulae becomes poorer in the case of structures with many heavy atoms for which the atomic amplitudes also contain an imaginary component (Shoemaker & Glauber, 1952). The experimental intensity measurement is made by a photo method or by direct recording (Avilov, 1979). In some cases the amplitudes $[\Phi_{hkl}]$ can be determined from dynamic scattering patterns – the bands of equal thickness from a wedge-shaped crystal (Cowley, 1981), or from rocking curves.

References

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International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 308-309