Structure factor

Steurer, W.; Haibach, T.

doi:10.1107/97809553602060000568

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

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International Tables for Crystallography (2006). Vol. B. ch. 4.6, pp. 500-501 | 1 | 2 |

Section 4.6.3.3.1.3. Structure factor

W. Steurer^a ^* and T. Haibach^a

^aLaboratory of Crystallography, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland
Correspondence e-mail: w.steurer@kristall.erdw.ethz.ch

4.6.3.3.1.3. Structure factor

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The structure factor of a periodic structure is defined as the Fourier transform of the density distribution $[\rho({\bf r})]$ of its unit cell (UC): $[F({\bf H}) = {\textstyle\int\limits_{\rm UC}} \rho({\bf r}) \exp(2\pi i{\bf H}\cdot {\bf r})\;\hbox{d}{\bf r}.]$ The same is valid in the case of the nD description of a quasiperiodic structure. The parallel- and perpendicular-space components are orthogonal to each other and can be separated. In the case of the 1D Fibonacci sequence, the Fourier transform of the parallel-space component of the electron-density distribution of a single atom gives the usual atomic scattering factor $[f({\bf H}^{\parallel})]$ . Parallel to $[x^{\perp}]$ , $[\rho({\bf r})]$ adopts values $[\neq 0]$ only within the interval $[- (1 + \tau)/]$ $[[2a^{*}(2 + \tau)] \leq {x}^{\perp} \leq (1 + \tau)/[2a^{*}(2 + \tau)]]$ and one obtains $[\eqalign{ F({\bf H}) &= f({\bf H}^{\parallel})[a^{*} (2 + \tau)]/(1 + \tau) \cr&\quad\times\textstyle\int\limits_{-(1 + \tau)/[2a^{*} (2 + \tau)]}^{+(1 + \tau)/[2a^{*} (2 + \tau)]} \exp(2\pi i{\bf H}^{\perp} \cdot x^{\perp})\;\hbox{d}x^{\perp}.}]$ The factor $[[a^{*}(2 + \tau)]/(1 + \tau)]$ results from the normalization of the structure factors to $[F({\bf 0}) = f(0)]$ . With $[\eqalign{{\bf H} &= h_{1}{\bf d}_{1}^{*} + h_{2}{\bf d}_{2}^{*} + h_{3}{\bf d}_{3}^{*} + h_{4}{\bf d}_{4}^{*}\cr &= h_{1}a_{1}^{*} \pmatrix{1\cr -\tau\cr 0\cr 0\cr} + h_{2}a_{1}^{*} \pmatrix{\tau\cr 1\cr 0\cr 0\cr} + h_{3}a_{3}^{*} \pmatrix{0\cr 0\cr 1\cr 0\cr} + h_{4}a_{4}^{*} \pmatrix{0\cr 0\cr 0\cr 1\cr}}]$ and $[{\bf H}^{\perp} = a_{1}^{*} (-\tau h_{1} + h_{2})]$ the integrand can be rewritten as $[\eqalign{ F({\bf H}) &= f({\bf H}^{\parallel}) [a^{*}(2 + \tau)]/(1 + \tau)\cr&\quad\times \textstyle{\int\limits_{- (1 + \tau)/[2a^{*}(2 + \tau)]}^{+ (1 + \tau)/[2a^{*}(2 + \tau)]}} \exp [2\pi i (- \tau h_{1} + h_{2}) x^{\perp}] \;\hbox{d}x^{\perp},}]$ yielding $[\eqalign{F({\bf H}) &= f({\bf H}^{\parallel}) (2 + \tau)/[2\pi i (-\tau h_{1} + h_{2})(1 + \tau)]\cr&\quad \times\exp [2\pi i(- \tau h_{1} + h_{2}) x^{\perp}] \big|_{- (1 + \tau)/[2a^{*}(2 + \tau)]}^{+ (1 + \tau)/[2a^{*}(2 + \tau)]}.\cr}]$ Using $[\sin x = (\hbox{e}^{ix} - \hbox{e}^{- ix})/2i]$ gives $[\eqalign{F({\bf H}) &= f({\bf H}^{\parallel}) (2 + \tau)/[\pi(- \tau h_{1} + h_{2})(1 + \tau)] \cr&\quad\times\sin[\pi (1 + \tau)(- \tau h_{1} + h_{2})]/(2 + \tau).}]$ Thus, the structure factor has the form of the function $[\sin (x)/x]$ with x a perpendicular reciprocal-space coordinate. The upper and lower limiting curves of this function are given by the hyperbolae $[\pm 1/x]$ (Fig. 4.6.3.6). The continuous shape of $[F({\bf H})]$ as a function of $[{\bf H}^{\perp}]$ allows the estimation of an overall temperature factor and atomic scattering factor for reflection-data normalization (compare Figs. 4.6.3.6 and 4.6.3.7).

Figure 4.6.3.7| top | pdf |

The structure factors $[F({\bf H})]$ of the Fibonacci chain decorated with aluminium atoms (U_overall = 0.005 Å²) as a function of the parallel (above) and the perpendicular (below) component of the diffraction vector. The short distance is $[\hbox{S} = 2.5\;\hbox{\AA}]$ , all structure factors within $[0 \leq |{\bf H}| \leq 2.5\;\hbox{\AA}^{-1}]$ have been calculated and normalized to [F(00) = 1] .

In the case of a 3D crystal structure which is quasiperiodic in one direction, the structure factor can be written in the form $[F({\bf H}) = {\textstyle\sum\limits_{k = 1}^{n}} \left[T_{k}({\bf H})f_{k}({\bf H}^{\parallel})g_{k}({\bf H}^{\perp}) \exp (2\pi i {\bf H} \cdot {\bf r}_{k})\right].]$ The sum runs over all n averaged hyperatoms in the 4D unit cell of the structure. The geometric form factor $[g_{k}({\bf H}^{\perp})]$ corresponds to the Fourier transform of the kth atomic surface, $[g_{k} ({\bf H}^{\perp}) = (1/A_{\rm UC}^{\perp}) {\textstyle\int\limits_{A_{k}}} \exp (2\pi i {\bf H}^{\perp} \cdot {\bf r}^{\perp})\ \hbox{d}{\bf r}^{\perp},]$ normalized to $[A_{\rm UC}^{\perp}]$ , the area of the 2D unit cell projected upon $[{\bf V}^{\perp}]$ , and $[A_{k}]$ , the area of the kth atomic surface.

The atomic temperature factor $[T_{k}({\bf H})]$ can also have perpendicular-space components. Assuming only harmonic (static or dynamic) displacements in parallel and perpendicular space one obtains, in analogy to the usual expression (Willis & Pryor, 1975), $[\eqalign{T_{k}({\bf H}) &= T_{k}({\bf H}^{\parallel},{\bf H}^{\perp})\cr& = \exp (-2\pi^{2}{\bf H}^{\| T} \langle {\bf u}_{i}^{\parallel} {\bf u}_{j}^{\| T}\rangle {\bf H}^{\parallel}) \exp (-2\pi^{2}{\bf H}^{\perp T} \langle {\bf u}_{i}^{\perp} {\bf u}_{j}^{\perp T}\rangle {\bf H}^{\perp}),\cr}]$ with $[\displaylines{ \langle {\bf u}_{i}^{\parallel} {\bf u}_{j}^{\| T}\rangle = \pmatrix{\langle {\bf u}_{1}^{\parallel 2}\rangle &\langle {\bf u}_{1}^{\parallel} \cdot {\bf u}_{2}^{\| T}\rangle &\langle {\bf u}_{1}^{\parallel} \cdot {\bf u}_{3}^{\| T}\rangle\cr \langle {\bf u}_{2}^{\parallel} \cdot {\bf u}_{1}^{\| T}\rangle &\langle {\bf u}_{2}^{\parallel 2}\rangle &\langle {\bf u}_{2}^{\parallel} \cdot {\bf u}_{3}^{\| T}\rangle\cr \langle {\bf u}_{3}^{\parallel} \cdot {\bf u}_{1}^{\| T}\rangle &\langle {\bf u}_{3}^{\parallel} \cdot {\bf u}_{2}^{\| T}\rangle &\langle {\bf u}_{3}^{\parallel 2}\rangle\cr}\cr\cr \hbox{ and } \langle {\bf u}_{i}^{\perp} {\bf u}_{j}^{\perp T}\rangle = \langle {\bf u}_{4}^{\perp}\rangle.}]$ The elements of the type $[\langle {\bf u}_{i} \cdot {\bf u}_{j}^{T}\rangle]$ represent the average values of the atomic displacements along the ith axis times the displacement along the jth axis on the V basis.

References

Willis, B. T. M. & Pryor, A. W. (1975). Thermal vibrations in crystallography. Cambridge University Press.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.6, pp. 500-501