Infogroup~metawiki

Nickname: blackbird

Entwickler der Science Software Infochannel und aktives Mitglied bei Wikipedia.

Programmbeschreibung: Infochannel (Grafiken, Diagramme und der Anhang werden noch eingefügt!)

Failed to parse (unknown function "\documentclass"): {\displaystyle \documentclass{article} \usepackage[ansinew]{inputenc} \usepackage[T1]{fontenc} \usepackage{ngerman} \pagestyle{headings} \begin{document} \textwidth15.5cm \textheight23cm \topmargin-10mm \oddsidemargin0mm \evensidemargin-4.5mm \title{\bf{INFOCHANNEL - A computer program to Convert energy dispersive high pressure powder X-ray diffraction data for Rietveld Analysis}} \author{\large {Dr. H.W.Neuling\thanks{Email:skeletor@physik.uni-paderborn.de}} \\ High Pressure Physics Laboratory \\ University of Paderborn \\ Warburger Str. 100 \\ D-4790 Paderborn} \maketitle \tableofcontents \listoffigures \newpage \section{Introduction} {\large Structure refinements using powder diffraction data taken with monochromatic X-rays are generally based on the Rietveld Method\cite{rv} and are widely applied in studies with samples at ambient pressures \cite{th}.With respect to high pressure, most of the recent diffraction studies with polycrystalline samples made use of the energy dispersive X-ray diffraction (EDXD) technique\cite{ho}.Apparently, systematic structure refinements based on such data makes it desirable to have a new computer program which converts them into conventional angle dispersive diffraction data for Rietveld analysis. \section{Experimental set-up} In a typical high pressure experiment with polychromatic radiation, the set-up consists of a conventional X-ray tube (using its white Bremsstrahlung) or a synchrotron radiation source, a primary collimating system, a diamond anvil cell (with a polycrystalline sample), a downstream slit system (defining the diffraction angle), an energy dispersive solid state detector (for collecting the diffracted intensities in a specified energy range), and finally electronics connected to a Computer with software for analysis of MCA data. \newpage \section{Energy of a diffracted beam} In order for X-rays to be diffracted from the hkl plane of a crystal, the Bragg equation has to be satisfied.In the case of angle dispersive methods, monochromatic radiation is used, and the Bragg condition is: \begin{equation} \lambda_0=2 d_{hkl}\sin\theta_{hkl} \end{equation} For the energy dispersive case, polychromatic radiation is used at a fixed diffraction angle $2\theta_{hkl}$, and the Bragg condition is: \begin{equation} E_{hkl}=\frac{6.199[keV \AA]}{d_{hkl}\sin\theta_0} \end{equation} \section{Intensity of a diffracted beam} For monochromatic radiation $(\lambda=\lambda_0)$, the integrated powder intensity is given by Buras \cite{bu} as \begin{equation} i(\theta)=j_{hkl} i_0(\lambda_0) \Delta\lambda_0 V N^2 |F_{hkl,\theta}|^2 \lambda^3_0 \frac{p}{4 \sin\theta}. \end{equation} For polychromatic radiation $(\theta=\theta_0)$, one gets \begin{equation} i(\lambda)=j_{hkl} i_0(\lambda) V N^2 |F_{hkl,\lambda}|^2 \lambda^4 \frac{p \cos\theta_0 \Delta\theta_0}{4 \sin^2\theta_0} \end{equation} where \begin{itemize} \item $j_{hkl}$ multiplicity factor for the hkl reflection, \item $i_0(\lambda)$ primary intensity per unit wavelength range, \item $\Delta\lambda_0$ spectral width of the incident beam, \item $\theta$ Bragg angle, \item $\Delta\theta_0$ angular width of the scattered beam, \item V effective crystal volume, \item N number of unit cells per unit volume, \item $F_{hkl}$ structure factor \\ \\and \item p polarization factor. \end{itemize} \subsection{Comparison between $i(\theta)$ and $i(E)$} Assuming that $i_0(\lambda)$ is a slowly varying function of $\lambda$ over a small width of each peak, the integration can be performed using the value of $i_0(\lambda)$ at the peak position \cite{ml}. Using \begin{equation} i_0(\lambda)=i_0(E)\left|\frac{dE}{d\lambda} \right|= i_0(E)\frac{hc}{\lambda^2}=i_0(E)\frac{E^2}{hc} \end{equation} formula (4) can then be rewritten as \begin{equation} i(E)= S(\theta_0) j_{hkl} i_0(E) \eta(E) E^{-2} A(E,\theta_0) |F_{hkl,E}|^2 \end{equation} Neglecting the effect of anormalous dispersion, the structure factors in formular (1) and (6) are equal. For the calculation of $i(\theta)$ from the measured intensities $i(E)$, we finally get the equation \begin{equation} i(\theta)= S^{\star} \frac{i(E) E^2 Lp(\theta)A(\theta)}{i_0(E) \eta(E) A(E,\theta_0)} \end{equation} with \begin{itemize} \item $S^\star$ scale factor, \item $Lp(\theta)$ Lorentz-polarization factor, \item $\eta(E)$ detector efficiency \\ \\ and \item $A(E,\theta_0)$ absorption factor. \end{itemize} \subsubsection{Calculation of $i_0(E)$} The intensity distribution for synchrotron radiation is proportional to the spectral distribution $N(h\nu)$ of the bending magnet radiation, which is given by \begin{equation} N(h\nu)=1.256\times 10^7 \gamma G_1(y) \end{equation} where $\gamma$, as usual, is $1957 \cdot E(GeV)$ and \begin{equation} G_1(y)=y \int^{\infty}_y K_{5/3}(t)dt=\left\{ \begin{array}{ccc} 2^{2/3} \Gamma \left(\frac{2}{3} \right) y^{(1/3)} & if & y \ll 1, \\ \sqrt{\frac{\pi}{2}} \sqrt{y} e^{-y} & if & y \gg 1, \end{array} \right. \end{equation} Here $K_{5/3}(t)$ is a modified Bessel function of the second kind, \begin{equation} \Gamma(x)=\int^{\infty}_0 e^{-t} t^{x-1}dt \end{equation} is the well known Gamma-function, and y is a dimensionless photon energy \begin{equation} y=\frac{h\nu}{h\nu_c} \end{equation} In this equation, y is defined in terms of the critical photon energy, \begin{equation} h\nu_c=\frac{3hc\gamma^3}{4\pi\rho} \end{equation} where $\rho$ is the bending radius. \\ A detailed discussion of the solutions of $G_1(y)$ for different branches of the energy spectrum, shown on the right hand side of equation (9), is given by Sokolov and Ternov\cite{so}. The general solution of the $G_1-function$ is shown in Fig.1, and the particular solution for synchrotron radiation of DORIS in Fig.2. \begin{figure}[bth] \unitlength1cm \begin{picture}(11.0,10.0) \end{picture} \par \caption{Spectral distribution function of bending magnet radiation} \end{figure} \begin{figure}[bth] \unitlength1cm \begin{picture}(11.0,10.0) \end{picture} \par \caption{Spectral distribution function on DORIS II} \end{figure} \subsubsection{Calculation of $\eta(E)$} The efficiency of a solid state detector depends upon the incident energy and on the material used as detector crystal. The effiency as a function of energy for a germanium detector is shown in Fig.3. In the energy range between 11.103keV (Ge-absorption edge) and 30keV\cite{kn}, it is represented in the form \begin{equation} \eta(E)= 0.4443+0.0185 \cdot E/keV \end{equation} \begin{figure}[bth] \unitlength1cm \begin{picture}(11.0,10.0) \end{picture} \par \caption{Efficiency function $\eta(E)$ for a Ge-detector} \end{figure} \subsubsection{Absorption correction on an energy scale} For the correction of absorption, the energy dependence of the linear scattering coefficient $\sigma_{K-N}$ (Compton+Rayleigh) and the linear photoelectric absorption coefficient $\tau$ have to be taken into account. The Klein-Nishina expression\cite{kl} can be used as a good approximation for the calculation of $\sigma_{K-N}$. \begin{eqnarray} \sigma_{K-N} & = & 2\pi\frac{e^4}{m^2 e^4}\left[ \frac{1+\alpha}{\alpha^2} \left\{ \frac{2(1+\alpha)}{1+2\alpha}- \frac{ln(1+2\alpha)}{\alpha} \right\}+ \cdots \right \nonumber\\ & &\mbox{}+\left \frac{ln(1+2\alpha)}{2\alpha}-\frac{1+3\alpha}{(1+2\alpha)^2} \right] \end{eqnarray} with $\alpha=\frac{E}{mc^2}$ \\ An empirical equation for the evaluation of $\tau$ is given by Viktoreen\cite{v1,v2,v3} \begin{equation} \frac{\tau}{\rho}=C_i \lambda^3-D_i \lambda^4 \end{equation} where $C_i,D_i$ are constants, which are tabulated for the elements with $Z \leq83$ in the International tables\cite{it}. For the linear total mass absorption coefficient, he gets \begin{equation} \frac{\mu}{\rho}=\frac{\tau}{\rho}+\frac{\sigma_{K-N}ZN_A}{A} \end{equation} with \begin{itemize} \item $\mu$ linear total absorption coefficient $[cm^{-1}]$, \item $\rho$ density $[g/cm^3]$, \item Z atomic number, \item $N_A$ Avogadro's number \\ \\ and \item A atomic weight [g/mole]. \end{itemize}\\ Finally, the absorption factor $A_i$ is given by the expression \begin{equation} A_i=e^{-\mu_i(E) t_i(\theta_0) f_p} \end{equation} where $t_i$ is the absorption length of the penetrated media and $f_p$ is the packing factor, which corresponds to the ratio of the density of a powder sample to that of the compact solid. According to Buras et al.\cite{ge}$f_p$ is close to 0.5. \begin{figure}[bth] \unitlength1cm \begin{picture}(11.0,10.0) \end{picture} \par \caption{Absorption factor as a function of radiation energy} \end{figure} Fig.4 shows the energy dependence of the absorption factor, calculated with the program INFOCHANNEL, for different materials. \subsubsection{Content of the scale factor $S^\star$} The scale factor contains the energy independent quantities in the intensity formula (7) $(e.g. $L(\theta_0), p(\theta_0))$, and the remaining constants for the conversion. \paragraph{Polarization of Synchrotron radiation} The linear polarization of the primary synchrotron beam is a complex expression, and is discussed in detail elsewhere\cite{so,ku}.Here we write for short \begin{equation} P_L=\frac{I_\parallel-I_\perp}{I_\parallel+I_\perp} \end{equation} where $I_\parallel$ and $I_\perp$ are the intensities of the parallel and perpendicular components of the synchrotron beam with respect to the plane defined by the primary and diffracted beam (diffraction plane). \\ It was previously outlined\cite{st} that the polarization factor for SR can be expressed as \begin{equation} p(\theta_0)=0.5\cdot\left(1+\cos^2(2\theta_0)-P_L \sin^2(2\theta_0) \right) \end{equation} For a diffraction experiment with the diffraction plane parallel to the principal plane of the storage ring, we have $I_\perp=0$, which gives $P_L=+1$ and consequently \begin{equation} p(\theta_0)=\cos^2(2\theta_0) \end{equation} On the other hand,when the diffraction plane is perpendicular to the storage ring plane, then $I_\parallel=0$, hence$P_L=-1$ and thus p is independent of the diffraction angle. \begin{equation} p(\theta_0)=1 \end{equation} \subsubsection{Evaluation of $Lp(\theta)$} For the case of a monochromator, the correction for Lorentz-polarization for Debye-Scherrer lines is given in the Intern. tables\cite{in} as \begin{equation} Lp(\theta)=\frac{1+\cos^2(2\theta)}{\sin^2\theta \cos(\theta)} \end{equation} \subsubsection{Evaluation of $A(\theta)$} In medium-grade powders $(\mu R_i \le 0.05$, where $R_i$ is an effective radius), the effect of differential absorption is such that, in order to calculate the intensity of a reflection, an absorption correction factor according to \cite{is} must be used, \begin{equation} A(\theta)=\frac{1}{V_i}\int^{V_i}_0 e^{\left\{-(\mu_i-\bar{\mu})R_i \right\}}dv \end{equation} where $\mu_i$ is the linear absorption coefficient for the particulate material, and $\bar{\mu}$ the mean linear absorption coefficient for the solid material (forming the specimen). \\ For spherical crystals, the absorption factor is constant in the range $0^{\circ}\leq\theta\leq90^{\circ}$. \newpage \section{Artefacts in energy dispersive X-ray scattering} Although we have not yet discussed the counting and storage of the pulses to form the familiar spectrum displayed by the energy dispersive system, it is appropriate at this point to go over the various artefacts and imperfections that can occur in it. \begin{figure}[bth] \unitlength1cm \begin{picture}(11.0,10.0) \end{picture} \par \caption{Artefacts in energy dispersive spectra} \end{figure} In Fig.5 are indicated a number of the artefacts that might result from a polyenergetic synchrotron radiation beam of photons entering a Ge-detector. Obviously, when a range of energies is coming in, the appearance of the spectrum is complicated, and some of the artefacts may not be seen easily; for instance, the presence of a spectral background due to Bremsstrahlung will mask small escape peaks and regions of partial pile-up.\\ The principal peak is primarily an ideal Gaussian shape, whose width is determined by the resolution of the system. On the low energy side, the tail that is shown results from slow charge collection in the detector. \subsection{Escape peaks} At an energy of exactly 9.875keV and 10.981keV below the main peak, there are two much smaller peaks. These are the escape peaks. They result when the incoming X-rays ionize a germanium atom by knocking out a K-shell electron, and the resulting X-ray photon is not reabsorbed by the detector.\\ The ratio of size of the escape peaks to the parent peak is a function of energy. The energy determines the average depth of penetration of the incident photon into the detector, and consequently the probability of escape of the generated Ge X-rays. It is useful to consider the general case in which incident photons may enter the detector at an angle which is not identical to the diffraction angle $2\theta$ in an EDXD experiment. If the incident angle is $A$ (zero for normal incidence), the average depth of penetration is reduced by the cosine of $A$. Based on the absorption coefficient for X-rays of energy E in germanium $\mu_{Ge}(E)$, the depth distribution of the excited germanium atoms (and consequently the distribution of generated X-rays) in terms of perpendicular distance in the detector z and the density $\rho_{Ge}$ becomes: \begin{equation} \Phi(\rho,z)=\mu_{Ge}(E)\rho_{Ge}\sec(A) e^{-\mu_{Ge}(E)\rho_{Ge} z \sec(A)} \end{equation} and the fraction $\epsilon$ of those which escape is: \begin{equation} \frac{1}{2} \omega \left( \frac{r-1}{r} \right) \left[ 1- \frac{\mu_{Ge}(K)}{\mu_{Ge}(E)} \cdot \cos(A) \cdot \ln \left( 1+ \frac{\mu_{Ge}(E)}{\mu_{Ge}(K)\cos(A)} \right) \right] \end{equation} The factor of one-half comes from the fact, that for ionizations near the front surface of the detector, half of the photons are emitted toward the surface and half into the body of the detector. The constants $\omega$ (fluorescence yield=0.508 for Ge\footnote{calculated with $\omega=e^{2.373 \ln(Z)-8.902}$ \cite{he}}), r (the K-shell absorption edge jump ratio=7.24 for Ge\footnote{calculated with $\frac{r-1}{r}=1.15265\cdotZ^{-0.083868}$ \cite{ru}}, and expressions for the germanium mass absorption coefficients of energy and for its own characteristic K-shell X-rays ($218.7 cm^{-1}$), can be combined into a single expression used to numerically calculate the escape peak fraction: \begin{equation} \epsilon= 0.2189-5.2 \cdot 10^{-5} E^{2.76} cos(A) ln \left(1+\frac{4197.73}{E^{2.76} cos(A)} \right) \end{equation} From this, the height of the escape peak relative to the parent peak can be calculated: \begin{equation} k=\frac{\epsilon}{1-\epsilon} \end{equation} Because this ratio is small, it is convenient to remove escape peaks by applying a channel-by-channel subtraction routine to the spectrum, starting at the high-energy end and working downward towards the Ge K-edge; if $N(I)$ is the number of counts in channel I (equivalent to E) and $IS$ is equivalent to the Ge $K_\alpha$ energy, then the correction is represented by \begin{equation} N(I-IS)=N(I-IS)-k \cdot N(I) \end{equation} \subsection{Fluorescence lines} Also shown in the "artefact" spectrum is a small peak exactly at 9.875keV. This represents Ge $K_{\alpha}$ X-ray fluorescence from the dead Ge-layer on the front surface of the detector. Because of the proximity of this layer to the region in which slow charge collection takes place, and because of the possibility that the energy loss process may produce some free electrons in the dead region, thus reducing the magnitude of the collected charge pulse, this peak is very asymmetric, with a broad low energy tail. In most real spectra, it occurs just at the discontinuity in the continuum background of the spectrum due to the absorption edge step of the germanium detector. Consequently it is hard to detect, and even harder to correct for.\\ Additionally, the strong fluorescence lines in the spectrum are due to excitation of characteristic X-rays in the sample material. \subsection{Sum peaks} At twice the energy above the principal peak, there is a small peak called the "sum peak". This represents two X-rays entering the detector so close together in time that their pulses are added together in the amplifier and are not distinguished by the pile-up rejection circuitry. Sum peaks can be found at all possible combinations of main peak energies (c.f. Fig.5). The size of the sum peaks is very dependent on count rate, and at extremely high rates, even triple or higher sum events can be found.\\ Below the sum peak, there sometimes is a broad region containing counts from pulses that were partially piled up, with one pulse riding on the leading or trailing edge of another. Ideally, a pile-up rejector should eliminate these partial pileups. \newpage \section{Program description} The program has been written in Turbo Pascal 6.0. The principal structure of INFOCHANNEL 1.0 is shown in Fig.6. \begin{figure}[bth] \unitlength1cm \begin{picture}(11.0,16.0) \end{picture} \par \caption{Flow chart of INFOCHANNEL 1.0} \end{figure} The computation starts with data files $\left ( channel,I(E) \right)$ in the Canberra S-100\cite{ca} or ASCII format and ends with step scan data $\left ( 2\theta,I(\theta \right )$ in an ASCII file which is compatible with PC-WYRIET\cite{sc}. The boxes with dashed lines in the flow chart represent procedures which have not yet been implemented in the program. In the first step, the channel numbers are converted into energies by using the calibration formula \begin{equation} E= A_e+B_e \cdot channel+C_e \cdot channel^2 \end{equation} where $A_e, B_e$ and $C_e$ are input parameters obtained by a least square fit routine of the S-100 program.\\ To convert $I(E)$ to $I(\theta)$, all quantities in formula (7) have to be calculated. To correct for absorption of the diamonds, the sample, the up- and downstream air, and the Be window of the port hole, formulas (14-17) have to be used. The input parameters required by the program, are the absorption lengths of the penetrated media, the diamond dimensions and the atomic number of the sample. For the calculation of the detector efficiency function, formula (13) has to be used. The primary intensity distribution of the synchrotron radiation is calculated by interpolation\cite{nw} between the values of the $G_1(y)-function$, stored in the ASCII file GFUNC.COR. These data are identical to those tabulated in the book\cite{ma} of Margaritondo. This method is much faster than the calculation of $G_1(y))$ using formula (9). It is necessary to input the machine parameters of the synchrotron.\\ Before continuing the intensity calculation, the energy scale has to be converted into the $2\theta$ scale with formula (2). $\lambda_0$ and $\theta_0$ are the required input parameters. The WYRIET program requires that the following calculations be added to the intensity correction:\\ The measured intensity has to multiplied with the Lorentz-polarization factor as defined in formula (22). \\ Step-scan data in $2\theta$ have to be calculated with an accuracy of 4 digits by truncation of the mean value of $2\theta$. The corresponding intensities will be evaluated by interpolation using Newtons method\cite{nw}.\\ Finally, the intensity scale factor has to be chosen in such way that the maximum intensities in the spectra do not exceed values of $10^5$. \newpage \addcontentsline{toc}{section}{7 \cdot References} \begin{thebibliography}{99} \bibitem{rv} H. M. Rietveld, J. Appl. Cryst 2, \underline{65} (1969). \bibitem{th} P. Thompson, D. E. Cox and J. B. Hastings, J. Appl. Cryst. \underline{20}, 79 (1987). \bibitem{ho} W. B. Holzapfel and P. G. Johannsen, High Pressure Science and Technology, three special volumes of High Pressure Research, \underline{3-5} (1990). \bibitem{bu} B.Buras, Handbook on Synchrotron Radiation, Vol. 1, p. 1015, ed. E. E. Koch, North-Holland Publishing Company (1983). \bibitem{ml} M. von Laue, Roentgenstrahl-Interferenzen, Akademische Verlagsgesellschaft, Frankfurt am Main (1960). \bibitem{so} A. A. Sokolov and I. M. Ternov, Synchrotron Radiation, ed. E. Schmutzer, Pergamon Press Oxford, p. 99 (1968). \bibitem{kn} G. F. Knoll, Radiation Detection and Measurement, Wyley \& sons New York (1979). \bibitem{kl} O.Klein and Y. Nishina, Z. Phys. \underline{52}, 853 (1929). \bibitem{v1} J. A. Viktoreen, J. Appl. Phys. \underline{19}, 855 (1948). \bibitem{v2} Idem Ibid, \underline{20}, 1141 (1949). \bibitem{v3} Idem Ibid, \underline{14}, 95 (1943). \bibitem{it} International Tables for X-Ray Crystallography, Vol. III, D. Reidel Publishing Company, p. 171 (1985). \bibitem{ge} B. Buras, L. Gerward, A. M. Glazer, M. Hidaka and J. Staun-Olsen, J. Appl. Cryst. \underline{12}, 531 (1979). \bibitem{ku} C. Kunz, Synchrotron Radiation, Topics in Current Physics, Vol. 10, Springer Verlag Heidelberg, p.9 (1979). \bibitem{st} B. Buras, J. Staun-Olsen, L. Gerward, G. Will and E. Hinze, J. Appl. Cryst. \underline{10}, 431 (1977). \bibitem{in} International Tables for X-Ray Crystallography Vol.II, Kluwer Academic Publishers, p. 266 (1989). \bibitem{is} Idem Ibid, Vol. III, D. Reidel Publishing Company, p.195 (1985). \bibitem{he} H. Yakowitz, R. L. Myklebust and K. F. J. Heinrich, Nat. Bur. of Stand. Tchnical Note 796, U. S. Dept. of Commerce (1973). \bibitem{ru} J. C. Russ, Fundamentals of Energy Dispersive X-ray Analysis, Butterworths Monographs in Materials London, p.83 (1984) \bibitem{ca} Canberra Industries Inc., System 100 User's Manual, p. 5.19 (1987). \bibitem{sc} J. Schneider, IUCr International workshop on the Rietveld method, Petten, Netherlands (1989). \bibitem{nw} W. Brauch, Programmierung mit Fortran, Teubner Verlag Stuttgart, p. 152 (1979). \bibitem{ma} G.Margaritondo, Introduction to Synchrotron Radiation, Oxford University Press, p. 246 (1988). \end{thebibliography} \newpage \begin{appendix} \section{Structure of Input-/Output-files} \subsection{GFUNC.COR} \subsection{Parameter file *.COP} \newpage \section{Description of PC WYRIET} \subsection{Rietveld Program DBW 3.2s (Version 8804)} \newpage \subsection{Input structure of DBW 3.2s} \newpage \subsection{Global parameters of inputfile FORT5} \newpage \subsection{Implemented reflection profile functions} \newpage \subsection{Output control flags of inputfile FORT5} \newpage \subsection{Quantities of refinement quality in FORT6} \end{appendix} \end{document} }