(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 737034, 21520]*) (*NotebookOutlinePosition[ 737824, 21547]*) (* CellTagsIndexPosition[ 737780, 21543]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Problem 4: Fast Fourier Transforms", "Section"], Cell[CellGroupData[{ Cell["Part A: Perfect Crystals", "Subsection"], Cell[TextData[{ "The diffraction pattern of a perfect crystal is a set of \[Delta]-function \ peaks at the reciprocal lattice points. Consider a 1-D crystal with atoms \ centered at", Cell[BoxData[ \(TraditionalForm\`\(\ R\_n\)\)]], " whose electron density forms Gaussians of width ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_n\)]], ": \[Rho](x)=", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = \[Placeholder]\)\%\[Placeholder] e \^\(\(-\((x - R\_n)\)\^2\)\/\(2 \[Sigma]\_n\^2\)\)\)]], ". Initially, let the lattice constant a=2.3 and \[Sigma] be 0.2. Start \ with a matrix for \[Rho] of 256 entries, spanning 32 lattice periods. The \ matrix RHO should be periodic too: make RHO(256) what RHO(0) would have \ been. The sum over n doesn't need to extend very far beyond the range of x. \ Calculate the Fourier transform ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&_\)]], "(k) using the FFT routine. Examine \[Rho](x) on the screen to make sure \ it looks like an array of atoms. Plot the power spectrum", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"\[VerticalSeparator]", RowBox[{ RowBox[{ FormBox[\(\[Rho]\&_\), "TraditionalForm"], "(", "k", ")"}], \(\[VerticalSeparator]\^2\)}]}]}], TraditionalForm]]], "over a range which includes the first four Bragg peaks. " }], "Text"], Cell["Definition of Constants:\t", "Text"], Cell[BoxData[ \(a = 2.3; \n\[Sigma] = 0.2; \nlatticeperiods = 32; \)], "Input"], Cell["\<\ Definition of a lattice with 256 elements. The reallattice corresponds to an \ actual lattice with spacing a.\ \>", "Text"], Cell[BoxData[ \(lattice = Table[i, {i, 256}]; \nlattice[\([256]\)] = lattice[\([1]\)]; \nreallattice = a*lattice; \)], "Input"], Cell["\<\ R[n_] is a function that defines the distance of the particular lattice from \ the origin of the lattice. Sumrange determines the extent of the summation \ over n for the electron density. 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When the plot is displayed in \ color, this is obvious, but in B&W it is difficult to distinguish the peak \ from the y - axis." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Part B: Diatomic Basis", "Subsection"], Cell[TextData[{ "If the unit cell of a crystal doubles in size, the reciprocal lattice \ divides in two. Let ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_n\)]], " be 0.1 if n is even and 0.3 if n is odd. Examine your real space \ density; for the same range of x (now 16 lattice periods) and the same other \ parameters, plot the power spectrum. " }], "Text"], Cell[TextData[ "All constants and variables are defined the same as in part a, but now \ \[Sigma] is a function of n and \n\[Rho][x_] takes this difference into \ account. 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