April 13, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 46April 13, 2005 Physics for Scientists&Engineers 2 2ReviewReview! The criterion for being able to separate two point objectsis called Rayleigh’s Criterion and is expressed as!R= sin"11.22#d$%&'()Resolved Barely Resolved Not ResolvedApril 13, 2005 Physics for Scientists&Engineers 2 3Review (2)Review (2)! With diffraction effects the intensity of the interference patternfrom double slits is given by! If the screen is placed a sufficiently large distance from the slits thenwe can writeI = Imaxcos2!sin""#$%&'(2 "=)a*sin+ !=)d*sin+!="ay#L and $="dy#LApril 13, 2005 Physics for Scientists&Engineers 2 4Review (3)Review (3)! A diffraction grating has a large number of slits, or rulings,placed very close together! To produce bright lines or constructive interference thispath length difference must be an integer multiple of thewavelength so! The values of m correspond to different bright lines! The dispersion describes the ability of a diffractiongrating to spread apart the various ordersd sin!= m" m = 0,1, 2,...( )D =!"!#=md cos" m = 1,2, 3,...( )April 13, 2005 Physics for Scientists&Engineers 2 5Resolving Power of a GratingResolving Power of a Grating! The resolving power R of a diffraction grating describes the ability ofthe diffraction grating to resolve closely spaced maxima, whichdepends on the width of each maximum! We define the power of a diffraction grating to resolve twowavelengths, !1 and !2, as! Thus to discuss the resolving power, we need an expression for thewidth of each maximum! The width of each maximum is defined by theposition of the first minimum on each side ofthe maximum! We can then define the half-width "hw of themaximum as the angle between the maximum and the first minimumR =!ave"! !ave=!1+!2( )/ 2 "!=!2#!1April 13, 2005 Physics for Scientists&Engineers 2 6Resolving Power of a Grating (2)Resolving Power of a Grating (2)! We base our argument our analysis of singleslit diffraction using the whole grating asthe single slit as shown! The angle of the first minimum for singleslit diffraction can be obtained where wesubstitute Nd for the slit width a! Because "hw is small, we can write! One can show that the width of the maxima for other orders is• ! is the angle corresponding to the maximum intensity for that orderNd sin!hw="!hw="Nd!hw="Nd cos!April 13, 2005 Physics for Scientists&Engineers 2 7Resolving Power of a Grating (3)Resolving Power of a Grating (3)! We can substitute "hw for #"! Which gives us! Note that the resolving power of a diffraction gratingdepends on the total number of rulings and the order!"!#=#Nd cos"!#=md cos"R =!"!= Nm !#!+!+ "!( )( )/ 2April 13, 2005 Physics for Scientists&Engineers 2 8X-Ray DiffractionX-Ray Diffraction! Wilhelm Röntgen discovered x-rays in the late 1800’s! These experiments suggested that x-rays were electromagnetic waveswith a wavelength of about 10-10 m! At about the same time, the study of crystalline solids suggested thatthe atoms of those solids were arranged in a regular repeating patternwith a spacing of about 10-10 m between the atoms! Putting these two ideas together, Max von Laue proposed in the early1900’s that a crystal could serve as a three dimensional diffractiongrating for x-rays! Von Laue and Friederich Knipping did the first x-ray diffractionexperiment that showed diffraction of x-rays by a crystal in 1912! Soon after Sir William Bragg and his son William Bragg derived Bragg’slaw and carried out a series of experiments involving x-ray diffractionfrom crystalsApril 13, 2005 Physics for Scientists&Engineers 2 9X-Ray Diffraction (2)X-Ray Diffraction (2)! Let’s assume that we have a cubic crystal as shown! Each atom in the lattice is a distance a away from the nextatom in all three directions! We can imagine various planes of atoms in this crystalApril 13, 2005 Physics for Scientists&Engineers 2 10X-Ray Diffraction (3)X-Ray Diffraction (3)! For example, the horizontal planes are composed of atomsspaced a distance a apart with the planes themselves beingspaced a distance a from each other! We can imagine x-rays incident on these planes and thatthe rows of atoms in the crystalline lattice can act like adiffraction grating! The x-rays can be thought of as scattering from the atomsApril 13, 2005 Physics for Scientists&Engineers 2 11X-Ray Diffraction (4)X-Ray Diffraction (4)! Interference effects are caused by path lengthdifferences! If we look at x-rays scattering off one plane, all the wavesremain in phase! However, if we consider adjacent planes, we can see belowthat the path length difference for the scattered x-raysfrom the two planes is! The criterion forconstructive interference is!x = !x1+ !x2= 2a sin"2a sin!= m" m = 0,1, 2,...( )April 13, 2005 Physics for Scientists&Engineers 2 12X-Ray Diffraction (5)X-Ray Diffraction (5)! Of course when x-rays are incident on a crystal, there canbe several different planes that can function as diffractiongratings! Some examples are illustrated below! These planes will not have the spacing a between the planesApril 13, 2005 Physics for Scientists&Engineers 2 13X-Ray Diffraction (6)X-Ray Diffraction (6)! To study the atomic structure of a substance using x-ray diffractionone can scatter x-rays parallel to the surface of a sample as shownbelow in a) or one can transmit the x-rays through the sample anddetect the x-rays on the opposite side of the sample and shown in b)April 13, 2005 Physics for Scientists&Engineers 2 14X-Ray Diffraction (7)X-Ray Diffraction (7)! For the parallel scattering method, the angle of incidence "should equal the angle of observation! For the transmission method, the observed angle is twicethe Bragg angle "! By measuring the intensity of the x-rays as a function of "one can determine details of the structure of the materialbeing studied! Modern particle accelerators such as the NationalSynchrotron Light Source at Brookhaven NationalLaboratory are used to produce high quality, intense beamsof x-rays to carry out material science
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