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Integrated intensity of a peak depends upon geometry...•Maximum intensity (Imax) scattered at each Bragg position proportional to 1 / sin θB•Intensity is large at small and small at large scattering angles•Breadth of each Bragg peak is proportional to 1 / cos θB•Breadth is small at small and large at large scattering angles•Integrated intensity of a Bragg reflection is therefore given by the product of these, i.e., Imax•B•Which is proportional to (1 / sin θB) • (1 / cos θB) or 1 / (2 sin θB) •Integrated intensity is greater at small and large scattering angles, less at intermediate scattering angles
Department of Materials Science & Engineering | UC BerkeleyFebruary 2, 201619MATERIALSSCIENCE& ENGINEERINGBerkeleyUNIVERSITY OFCALIFORNIAIntegrated Intensity (2)Integrated intensity of a diffraction peak depends upon the number of crystals in correct orientation (from available random distribution)•The ratio of the number of properly-oriented crystals ∆𝑁to the total number of crystals 𝑁in the powder is obtained by the areal fraction of scattering (area of strip between 2 cones to area of full sphere)•Number of favorably-oriented crystals is proportional to cos 𝜃?and is very small for scattering in backwards direction∆𝑁𝑁=𝑟∆𝜃 ∙ 2𝜋𝑟 sin90° − 𝜃?4𝜋𝑟2∆𝑁𝑁=∆𝜃 cos 𝜃?2
Department of Materials Science & Engineering | UC BerkeleyFebruary 2, 201620MATERIALSSCIENCE& ENGINEERINGBerkeleyUNIVERSITY OFCALIFORNIAIntegrated Intensity (3)Proper comparison of integrated intensities requires a comparison of integrated intensity per unit length of a diffraction line…•Comparison of total diffracted intensity in a single cone with that in another cone is not always possible•More of the cone is intercepted in the forward-scattered or back-scattered regime•Length of any given diffraction line in powder method is 2𝜋𝑅 sin 2𝜃?, where Ris the radius of the camera•Relative intensity per unit length of line is proportional to 1sin 2𝜃𝐵
Department of Materials Science & Engineering | UC BerkeleyFebruary 2, 201621MATERIALSSCIENCE& ENGINEERINGBerkeleyUNIVERSITY OFCALIFORNIALorentz Factor / Lorentz-Polarization Factor Combining these three geometrical factors affecting the total integrated intensity of an X-ray diffraction peak Lorentz factor is defined??𝑟???? 𝐹𝑎???𝑟 =1sin 2𝜃?cos 𝜃?1sin 2𝜃?=cos 𝜃?sin22𝜃?=14 sin22𝜃?cos 𝜃?•This is combined with the polarization factor 121 + cos22𝜃?•Lorentz-Polarization Factor = 1+cos22𝜃𝐵sin22𝜃𝐵cos 𝜃𝐵•Describes how geometrical factorsdecrease the intensity of reflectionsat intermediate angles compared to those in forward or backward direction04590Bragg Angle (degrees)01020304050L-P factor
Department of Materials Science & Engineering | UC BerkeleyFebruary 2, 201622MATERIALSSCIENCE& ENGINEERINGBerkeleyUNIVERSITY OFCALIFORNIAAbsorption Factor•