Ch2-_Diffraction.ipynb

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    "<font size = \"5\"> **Chapter 2: [Diffraction](Ch2-Diffraction.ipynb)** </font>\n",
    "\n",
    "\n",
    "<hr style=\"height:1px;border-top:4px solid #FF8200\" />\n",
    "\n",
    "part of \n",
    "\n",
    "\n",
    "<font size = \"5\"> **[Analysis of Transmission Electron Microscope Data](_Analysis_of_Transmission_Electron_Microscope_Data.ipynb)**</font>\n",
    "\n",
    "\n",
    "by Gerd Duscher, 2019\n",
    "\n",
    "Microscopy Facilities<br>\n",
    "Joint Institute of Advanced Materials<br>\n",
    "The University of Tennessee, Knoxville\n",
    "\n",
    "Model based analysis and quantification of data acquired with transmission electron microscopes\n",
    "part of\n",
    "\n",
    "The diffraction chapter has the following sections:\n",
    "## Content\n",
    "The Diffraction chapter has the following sections:\n",
    "* [The Electron](CH2-Electron.ipynb)\n",
    "* [Atomic Form Factor](Ch2-Atomic_Form_Factor.ipynb)\n",
    "* [Basic Crystallography](Ch2-Basic_Crystallography.ipynb)\n",
    "    * [Plot Unit Cell with Other Packages](Plot_UnitCell.ipynb)\n",
    "    * [Fourier Transform Laboratory](FourierTransformLaboratory.ipynb)\n",
    "* [Structure Factors](CH2-Structure_Factors.ipynb)\n",
    "* [Analyzing Ring Diffraction Pattern](CH2-Diffraction-Rings.ipynb)\n",
    "* [Kinematic Scattering Geometry](CH2-Kinematic_Scattering_Geometry.ipynb)\n",
    "* [Plotting of Diffraction Pattern](CH2-Plotting_Diffraction_Pattern.ipynb)\n",
    "* [Analyzing Spot Diffraction Pattern](CH2-Spot_Diffraction_Pattern.ipynb)\n",
    "* [Kikuchi Lines](CH2-Kikuchi_Lines.ipynb)\n",
    "* [HOLZ Lines](HOLZ_Lines.ipynb)\n",
    "* [Lattice Determination with HOLZ](HOLZ_Example.ipynb) \n",
    "* [Kinematic Diffraction with KinsCat](Kinematic_Diffraction_with_KinsCat.ipynb)\n",
    "\n",
    "# Introduction\n",
    "\n",
    "## Basics\n",
    "\n",
    "- Diffraction is the direct result of the interaction (with or without\n",
    "energy transfer) of electrons and matter.\n",
    "- Kinematic diffraction theory describes only the Bragg angles (the position of the Bragg reflections) but not the intensity in a real\n",
    "diffraction pattern.\n",
    "- Dynamic theory is responsible for the intensity variation of the Bragg reflections\n",
    "\n",
    "## Diffraction and Imaging\n",
    "\n",
    "- To achieve an image from a diffraction pattern only a **Fourier Transformation** of parts of the diffraction pattern is needed.\n",
    "- Any image in a TEM can be described as **Fourier Filtering**, because we select the beams that form the images. The knowledge of which and how many diffracted beams contribute to the image formation is crucial.\n",
    "- Because the intensity of selected diffracted beams is necessary to calculate image intensities, dynamic diffraction theory is necessary.\n",
    "- Understanding difraction theory of electrons is at the core of the analysis of TEM data.\n",
    "\n",
    "\n",
    "According to Fourier-Optical view of the TEM we first form a diffraction pattern (in Fourier space) and after another (inverse) Fourier\n",
    "Transformation of parts of this diffraction pattern we are back in real space and can observe an image.\n",
    "\n",
    "## Dynamic and Kinematic Theory\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "- Kinematic Theory is based on a single scattering event per electron\n",
    "- Kinematic theory is used in neutron and X-ray diffraction almost exclusively.\n",
    "\n",
    "- Dynamic theory incorporates multi scattering events.\n",
    "- Dynamic theory results in Rocking Curves of intensities of diffracted beams with sample thickness.\n",
    "- Dynamic theory can analytically be solved only for the two beam case, strangely the basis for conventional TEM.\n",
    "\n"
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    "## Diffraction and Scattering\n",
    "\n",
    "Electrons can be viewed as particles and/or as waves.\n",
    "\n",
    "### Particle-wave dualism:\n",
    "\n",
    " <table style=\"width:80%\">\n",
    "  <tr>\n",
    "    <th>Scattering</th>\n",
    "    <th>$\\leftrightarrow$</th>\n",
    "    <th>Diffraction</th>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td>Particle picture</td>\n",
    "    <td>$\\leftrightarrow$</td>\n",
    "    <td>Wave picture</td>\n",
    "  </tr>\n",
    " \n",
    "</table> \n",
    "\n",
    "We will switch back and forth between these two pictures, depending on which is mathematically easier to express.\n",
    "\n",
    "\n",
    "## Kinematic Diffraction Theory Buzz Words\n",
    "\n",
    "The following terms will be important in Kinematic Theory:\n",
    "\n",
    " <table style=\"width:80%\">\n",
    " \n",
    "  <tr>\n",
    "    <td>f</td>\n",
    "    <td> atomic scattering factor</td>\n",
    "    <td>scattering strength of an atom </td>\n",
    "        \n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td>F</td>\n",
    "    <td>form factor</td>\n",
    "      <td>combination of symmetrry and atomic scattering factor</td>\n",
    "  </tr>\n",
    "    <tr>\n",
    "    <td></td>\n",
    "    <td>forbidden reflection</td>\n",
    "      <td>a direct result of the form factor</td>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td>$\\sigma$</td>\n",
    "    <td>cross-section</td>\n",
    "      <td>scattering probability expressed as an effective area </td>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td>$\\frac{\\partial \\sigma}{\\partial \\Omega}$</td>\n",
    "    <td>cross-section</td>\n",
    "    <td> scattering probability in a solid angle </td>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td>$\\lambda$</td>\n",
    "    <td>mean free path</td>\n",
    "    <td> scattering probability expressed as a path-length between two scattering events</td>\n",
    "  </tr> \n",
    "</table> \n",
    "\n",
    "We will start discussiong this terms in the [Atomic Form Factor](Ch2-Atomic_Form_Factor.ipynb) and following notebooks"
   ]
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    "## Navigation\n",
    "### Back:  Chapter 1: [Introduction](CH1-Introduction.ipynb)\n",
    "### Next:  Chapter 3: [Imaging](CH3-Imaging.ipynb)\n",
    "\n",
    "### List of Content: [Front](_Analysis_of_Transmission_Electron_Microscope_Data.ipynb)"
   ]
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