Integrated intensity of a peak depends upon geometry

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Integrated intensity of a peak depends upon geometry... Maximum intensity (I max ) scattered at each Bragg position proportional to 1 / sin θ B Intensity is large at small and small at large scattering angles Breadth of each Bragg peak is proportional to 1 / cos θ B 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. , I max •B Which is proportional to (1 / sin θ B ) • (1 / cos θ B ) or 1 / (2 sin θ B ) Integrated intensity is greater at small and large scattering angles, less at intermediate scattering angles
Department of Materials Science & Engineering | UC Berkeley February 2, 2016 19 M ATERIALS S CIENCE & E NGINEERING Berkeley U NIVERSITY OF C ALIFORNIA Integrated 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 ∆𝑁 to the total number of crystals 𝑁 in 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 ∆𝑁 𝑁 = 𝑟∆𝜃 ∙ 2𝜋𝑟 sin 90° − 𝜃 ? 4𝜋𝑟 2 ∆𝑁 𝑁 = ∆𝜃 cos 𝜃 ? 2
Department of Materials Science & Engineering | UC Berkeley February 2, 2016 20 M ATERIALS S CIENCE & E NGINEERING Berkeley U NIVERSITY OF C ALIFORNIA Integrated 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 R is the radius of the camera Relative intensity per unit length of line is proportional to 1 sin 2𝜃 𝐵
Department of Materials Science & Engineering | UC Berkeley February 2, 2016 21 M ATERIALS S CIENCE & E NGINEERING Berkeley U NIVERSITY OF C ALIFORNIA Lorentz 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 = 1 sin 2𝜃 ? cos 𝜃 ? 1 sin 2𝜃 ? = cos 𝜃 ? sin 2 2𝜃 ? = 1 4 sin 2 2𝜃 ? cos 𝜃 ? This is combined with the polarization factor 1 2 1 + cos 2 2𝜃 ? Lorentz-Polarization Factor = 1+cos 2 2𝜃 𝐵 sin 2 2𝜃 𝐵 cos 𝜃 𝐵 Describes how geometrical factors decrease the intensity of reflections at intermediate angles compared to those in forward or backward direction 0 45 90 Bragg Angle (degrees) 0 10 20 30 40 50 L-P factor
Department of Materials Science & Engineering | UC Berkeley February 2, 2016 22 M ATERIALS S CIENCE & E NGINEERING Berkeley U NIVERSITY OF C ALIFORNIA Absorption Factor