Review The criterion for being able to separate two point objects is called Rayleigh s Criterion and is expressed as 1 22 R sin 1 d Physics for Scientists Engineers 2 Spring Semester 2005 Lecture 46 Resolved April 13 2005 Physics for Scientists Engineers 2 1 April 13 2005 A diffraction grating has a large number of slits or rulings placed very close together a d sin sin If the screen is placed a sufficiently large distance from the slits then we can write 2 Review 3 With diffraction effects the intensity of the interference pattern from double slits is given by 2 Not Resolved Physics for Scientists Engineers 2 Review 2 sin I I max cos 2 Barely Resolved ay dy and L L To produce bright lines or constructive interference this path length difference must be an integer multiple of the wavelength so d sin m m 0 1 2 The values of m correspond to different bright lines The dispersion describes the ability of a diffraction grating to spread apart the various orders D April 13 2005 Physics for Scientists Engineers 2 3 April 13 2005 m d cos m 1 2 3 Physics for Scientists Engineers 2 4 Resolving Power of a Grating Resolving Power of a Grating 2 We base our argument our analysis of single slit diffraction using the whole grating as the single slit as shown The resolving power R of a diffraction grating describes the ability of the diffraction grating to resolve closely spaced maxima which depends on the width of each maximum The angle of the first minimum for single slit diffraction can be obtained where we substitute Nd for the slit width a We define the power of a diffraction grating to resolve two wavelengths 1 and 2 as R ave Nd sin hw ave 1 2 2 2 1 Because hw is small we can write Thus to discuss the resolving power we need an expression for the width of each maximum hw The width of each maximum is defined by the position of the first minimum on each side of the maximum One can show that the width of the maxima for other orders is hw We can then define the half width hw of the maximum as the angle between the maximum and the first minimum April 13 2005 Physics for Scientists Engineers 2 5 April 13 2005 Physics for Scientists Engineers 2 6 X Ray Diffraction Wilhelm R ntgen discovered x rays in the late 1800 s We can substitute hw for These experiments suggested that x rays were electromagnetic waves with a wavelength of about 10 10 m m Nd cos d cos At about the same time the study of crystalline solids suggested that the atoms of those solids were arranged in a regular repeating pattern with a spacing of about 10 10 m between the atoms Which gives us Nm Nd cos is the angle corresponding to the maximum intensity for that order Resolving Power of a Grating 3 R Nd 2 Putting these two ideas together Max von Laue proposed in the early 1900 s that a crystal could serve as a three dimensional diffraction grating for x rays Note that the resolving power of a diffraction grating depends on the total number of rulings and the order Von Laue and Friederich Knipping did the first x ray diffraction experiment that showed diffraction of x rays by a crystal in 1912 Soon after Sir William Bragg and his son William Bragg derived Bragg s law and carried out a series of experiments involving x ray diffraction from crystals April 13 2005 Physics for Scientists Engineers 2 7 April 13 2005 Physics for Scientists Engineers 2 8 X Ray Diffraction 2 X Ray Diffraction 3 Let s assume that we have a cubic crystal as shown For example the horizontal planes are composed of atoms spaced a distance a apart with the planes themselves being spaced a distance a from each other We can imagine x rays incident on these planes and that the rows of atoms in the crystalline lattice can act like a diffraction grating The x rays can be thought of as scattering from the atoms Each atom in the lattice is a distance a away from the next atom in all three directions We can imagine various planes of atoms in this crystal April 13 2005 Physics for Scientists Engineers 2 9 April 13 2005 X Ray Diffraction 4 Physics for Scientists Engineers 2 10 X Ray Diffraction 5 Interference effects are caused by path length differences Of course when x rays are incident on a crystal there can be several different planes that can function as diffraction gratings If we look at x rays scattering off one plane all the waves remain in phase Some examples are illustrated below These planes will not have the spacing a between the planes However if we consider adjacent planes we can see below that the path length difference for the scattered x rays from the two planes is x x1 x2 2a sin The criterion for constructive interference is 2a sin m April 13 2005 m 0 1 2 Physics for Scientists Engineers 2 11 April 13 2005 Physics for Scientists Engineers 2 12 X Ray Diffraction 6 X Ray Diffraction 7 To study the atomic structure of a substance using x ray diffraction one can scatter x rays parallel to the surface of a sample as shown below in a or one can transmit the x rays through the sample and detect the x rays on the opposite side of the sample and shown in b For the parallel scattering method the angle of incidence should equal the angle of observation For the transmission method the observed angle is twice the Bragg angle By measuring the intensity of the x rays as a function of one can determine details of the structure of the material being studied Modern particle accelerators such as the National Synchrotron Light Source at Brookhaven National Laboratory are used to produce high quality intense beams of x rays to carry out material science research April 13 2005 Physics for Scientists Engineers 2 13 April 13 2005 Physics for Scientists Engineers 2 14
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