Microsoft Power Point x Ray Diffraction

8/2/2019 Microsoft Power Point x Ray Diffraction

1/19

XX--ray Diffractionray DiffractionThe BasicsThe Basics

Followed by a few examples ofFollowed by a few examples ofData AnalysisData Analysis

bybyWesley TennysonWesley Tennyson

NanoLab/NSF NUE/BummNanoLab/NSF NUE/Bumm


2/19

XX--ray Diffractionray DiffractionBraggs LawBraggs Law

Lattice ConstantsLattice ConstantsLaue ConditionsLaue Conditions


-- 22ScanScanScherrers FormulaScherrers Formula

Data Analysis ExamplesData Analysis Examples


3/19

Braggs Lawn = 2 dsin

Constructive interference only occurs for certain s

correlating to a (hkl) plane, specifically when the path

difference is equal to n wavelengths.


4/19

Bragg conditionsBragg conditionsThe diffraction condition can be written in vectorThe diffraction condition can be written in vector

formform

22kkGG ++ GG22 = 0= 0


kk -- is the incident wave vectoris the incident wave vector

kk -- is the reflected wave vectoris the reflected wave vector

GG -- is a reciprocal lattice vector such that whereis a reciprocal lattice vector such that where

GG == kk == kk -- kk

the diffraction condition is metthe diffraction condition is met


5/19

Lattice ConstantsLattice ConstantsThe distance between planes of atoms isThe distance between planes of atoms is

dd((hklhkl) = 2) = 2/ |/ |GG||

SinceSince GG can be written ascan be written as


GG == aa 11++ 22 ++ 33Substitute inSubstitute in GG

dd((hklhkl) =) = aa/ (/ (hh22 ++ kk22 ++ ll22))(1/2)(1/2)

OrOr

aa == dd** ((hh22 ++ kk22 ++ ll22))(1/2)(1/2)

aa is the spacing between nearest neighborsis the spacing between nearest neighbors


6/19

Laue ConditionsLaue Conditionsaa11kk = 2= 211 aa22kk = 2= 222

aa33kk = 2= 233


Each of the above describes a cone in reciprocalEach of the above describes a cone in reciprocalspace about the lattice vectorsspace about the lattice vectors aa11,, aa22, and, and aa33..

thethe ii are integersare integers

When a reciprocal lattice point intersects this cone theWhen a reciprocal lattice point intersects this cone thediffraction condition is met, this is generally calleddiffraction condition is met, this is generally called

the Ewald sphere.the Ewald sphere.


7/19

Summary of Bragg & LaueSummary of Bragg & LaueWhen a diffractionWhen a diffraction

condition is met there cancondition is met there canbe a reflected Xbe a reflected X--rayray Extra atoms in the basis canExtra atoms in the basis can

sin2

nd =


Three variablesThree variables ,, , and, and dd is knownis known is measured in theis measured in the

experiment (2experiment (2)) d is calculatedd is calculated

From the planes (From the planes (hklhkl)) aais calculatedis calculated

222

lkhda ++=


8/19

-- 22ScanScanTheThe -- 22scan maintains these angles with thescan maintains these angles with the

sample, detector and Xsample, detector and X--ray sourceray source

Normal to surface

NanoLab/NSF NUE/BummNanoLab/NSF NUE/BummOnly planes of atoms that share this normal will be seen in the - 2Scan


9/19

-- 22ScanScanThe incident XThe incident X--rays may reflect in many directionsrays may reflect in many directions

but will only be measured at one location so webut will only be measured at one location so wewill require that:will require that:


Angle of incidence (Angle of incidence (ii) = Angle of reflection () = Angle of reflection (rr))

This is done by moving the detector twice as fastThis is done by moving the detector twice as fastinin as the source. So, only whereas the source. So, only where ii == rr is theis theintensity of the reflect wave (counts of photons)intensity of the reflect wave (counts of photons)measured.measured.


10/19

-- 22ScanScan



11/19

Smaller Crystals Produce Broader XRD PeaksSmaller Crystals Produce Broader XRD Peaks


12/19

Scherrers FormulaScherrers Formula

BcosB

Kt

=

t= thickness of crystalliteK= constant dependent on crystallite shape (0.89)= x-ray wavelengthB= FWHM (full width at half max) or integral breadthB = Bragg Angle


13/19

Scherrers FormulaScherrers FormulaWhat isWhat is BB??

BB= (2= (2High)High) (2(2Low)Low)

Peak

BBis the difference inis the difference inanglesangles at half max

2 high

Noise

2

low


14/19

When to Use Scherrers FormulaWhen to Use Scherrers Formula

Crystallite size


15/19

Data AnalysisData Analysis

Plot the data (2Plot the data (2vs. Counts)vs. Counts)Determine the Bragg Angles for the peaksDetermine the Bragg Angles for the peaks

CalculateCalculate ddandand aafor each peakfor each peak Apply Scherrers Formula to the peaksApply Scherrers Formula to the peaks


16/19

Bragg ExampleBragg Example


17/19

Bragg ExampleBragg Exampledd== / (2 Sin/ (2 Sin BB)) = 1.54 = 1.54

== 1.541.54 / ( 2 * Sin ( 38.3 / 2 ) ) / ( 2 * Sin ( 38.3 / 2 ) )= 2.35 = 2.35

Simple Right!Simple Right!


18/19

Scherrers ExampleScherrers ExampleAu Foil

98.25 (400)7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

95 95.5 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 101 101.5 102

2 Theta

Counts


19/19

Scherrers ExampleScherrers Example

BB

t

cos

89.0

=

tt = 0.89*= 0.89*/ (/ (BBCosCos BB)) = 1.54 = 1.54

= 0.89*= 0.89*1.541.54 / ( 0.00174 * Cos (98.25/ 2 ) )/ ( 0.00174 * Cos (98.25/ 2 ) )= 1200 = 1200

BB= (98.3= (98.3 -- 98.2)*98.2)*/180 = 0.00174/180 = 0.00174

Simple Right!Simple Right!

Documents

Microsoft Power Point x Ray Diffraction