3 Bragg’s Lawnλ = 2 d sin θConstructive interference only occurs for certain θ’s correlating to a (hkl) plane, specifically when the path difference is equal to n wavelengths.
4 the diffraction condition is met Bragg condition’sThe diffraction condition can be written in vector form2k∙G + G2 = 0k - is the incident wave vectork’ - is the reflected wave vectorG - is a reciprocal lattice vector such that whereG = ∆k = k - k’the diffraction condition is metNanoLab/NSF NUE/Bumm
5 Lattice Constants The distance between planes of atoms is d(hkl) = 2π / |G|Since G can be written asG = 2π/a (h*b1+ k*b2 +l*b3)Substitute in Gd(hkl) = a / (h2 + k2 + l2)(1/2)Ora = d * (h2 + k2 + l2)(1/2)a is the spacing between nearest neighborsNanoLab/NSF NUE/Bumm
6 Laue Conditions a1∙∆k = 2πυ1 a2∙∆k = 2πυ2 a3∙∆k = 2πυ3 Each of the above describes a cone in reciprocal space about the lattice vectors a1, a2, and a3.the υi are integersWhen a reciprocal lattice point intersects this cone the diffraction condition is met, this is generally called the Ewald sphere.NanoLab/NSF NUE/Bumm
7 Summary of Bragg & LaueWhen a diffraction condition is met there can be a reflected X-rayExtra atoms in the basis can suppress reflectionsThree variables λ, θ, and dλ is knownθ is measured in the experiment (2θ)d is calculatedFrom the planes (hkl)a is calculatedNanoLab/NSF NUE/Bumm
8 θ - 2θ ScanThe θ - 2θ scan maintains these angles with the sample, detector and X-ray sourceNormal to surfaceOnly planes of atoms that share this normal will be seen in the θ - 2θ ScanNanoLab/NSF NUE/Bumm
9 Angle of incidence (θi) = Angle of reflection (θr) θ - 2θ ScanThe incident X-rays may reflect in many directions but will only be measured at one location so we will require that:Angle of incidence (θi) = Angle of reflection (θr)This is done by moving the detector twice as fast in θ as the source. So, only where θi = θr is the intensity of the reflect wave (counts of photons) measured.NanoLab/NSF NUE/Bumm
12 Scherrer’s Formula t = thickness of crystallite K = constant dependent on crystallite shape (0.89)l = x-ray wavelengthB = FWHM (full width at half max) or integral breadthqB = Bragg Angle
13 Scherrer’s Formula What is B? B = (2θ High) – (2θ Low) B is the difference in angles at half maxPeak2θ low2θ highNoise
14 When to Use Scherrer’s Formula Crystallite size <1000 ÅPeak broadening by other factorsCauses of broadeningSizeStrainInstrumentIf breadth consistent for each peak then assured broadening due to crystallite sizeK depends on definition of t and BWithin 20%-30% accuracy at bestSherrer’s Formula ReferencesCorman, D. Scherrer’s Formula: Using XRD to Determine Average Diameter of Nanocrystals.
15 Data Analysis Plot the data (2θ vs. Counts) Determine the Bragg Angles for the peaksCalculate d and a for each peakApply Scherrer’s Formula to the peaks