9/13/2013. Diffraction. Diffraction. Diffraction. Diffraction. Diffraction. Diffraction of Visible Light

Size: px

Start display at page:

Download "9/13/2013. Diffraction. Diffraction. Diffraction. Diffraction. Diffraction. Diffraction of Visible Light"

Lynn Richards
6 years ago
Views:

$iffraction grating Joseph von$ Fraunhofer German 80 slits new

1 scattering of raiation by an object observe an escribe over 300 years ago illustrate with a iffraction grating Joseph von Fraunhofer German 80 slits new wavefront constructive interference exact pattern epens on wavelength () an istance () between slits wave of Visible Light secon-orer works if of raiation cos h h integer

$of X-rays in 90, it was not certain whether s were waves German (879-960) of X-rays first photograph of iffraction by crystal of us 4 Max von Laue suggeste that if crystals were perioic, with$ $cos c l ( 0 0 ) ( 0 0 ) ( 0 0 0 ) ( 0 0 ) ( 0 0 ) improve Laue iffraction photo of us 4 Frierich an Knipping crystal is stationary observe iffraction angle,, where iffraction cones intersect note: in$ $-D, iffraction from a point in -D, iffraction from a line in 3-D, iffraction from a (Miller) plane also in 9, William Lawrence ragg an his father William Henry ragg in phase A in phase$

2 of X-rays in 90, it was not certain whether s were waves German ( ) of X-rays first photograph of iffraction by crystal of us 4 Max von Laue suggeste that if crystals were perioic, with interatomic istances of ~ Å, they might act as iffraction gratings towars waves German 9 base on suggestion, Walther Frierich an aul Knipping emonstrate this using a crystal of us 4 (off the shelf) experimental setup use by Knipping an Frierich signe by Max von Laue Laue if polychromatic raiation use Laue Equations for a 3-D crystal, replace with cell imensions a, b, an c: a cos a h b cos b k c cos c l ( 0 0 ) ( 0 0 ) ( ) ( 0 0 ) ( 0 0 ) improve Laue iffraction photo of us 4 Frierich an Knipping crystal is stationary observe iffraction angle,, where iffraction cones intersect note: in -D, iffraction from a point in -D, iffraction from a line in 3-D, iffraction from a (Miller) plane also in 9, William Lawrence ragg an his father William Henry ragg in phase A in phase semi-transparent mirrors ritish (890-97) ritish (86-94) n wave in phase after reflection only if: A + is an integer multiple of note similarity of iffraction with reflection from a plane mirror A ; A n sin A/; A sin sin n

3 reflection conitions epen on,, an raggs Law says nothing about absolute intensities, just where maxima are foun reflecte beam eviates from irect beam by sin n still vali: all e ensity is not on the plane all e ensity has some contribution to all planes reflection is from regions of e ensity: atoms which act as points of Laue iffraction all e ensity is not on the plane still vali: all e ensity has some contribution to all planes it is variations in contribution to a reflection that accounts for iffering intensities of reflections from various planes; allows for structure etermination Reflection from e ensity off the plane is slightly out of phase. Intensity is less. X-rays reflect from all Miller planes in the unit cell; crystallographic ata often referre to as reflections since monochromatic raiation is use, the eviation of a reflection epens on the interplanar spacing, 3

4 Example sin n in crystallography, n for all reflections higher orer reflections (n, 3, etc) are the same as st orer reflections from parallel planes which are an n multiple of the original plane n orer reflection from ( 0 0) same as st orer reflection from ( 0 0) y z cubic unit cell: a 4.00 Å uk raiation:.54 Å x 4.00 st orer from ( 0 0).º.00 Å for the n orer reflection from ( 0 0)? sin n ; sin (n/) sin ( ((.54 Å)) / ((4.00 Å) ).7 o for the st orer reflection from ( 0 0)? θ 45.4º sin ( ((.54 Å)) / ((.00 Å) ).7 o Reciprocal Space sin n sin n sin inversely proportional to large means compresse iffraction an vise versa a c select an origin monoclinic a 0.75 Å b.00 Å c.00 Å 05 o to get a irect relationship, instructive to efine: looking own b axis Reciprocal Space Reciprocal Lattice Direct Lattice star notation: x x* * 0 0 * 0 0 raw a vector from origin to plane, with a length of / select an arbitrary origin ( 0 0) planes (0 0 ) planes 4

5 * 0 0 * 0 ( 0 ) planes (0 0 ) planes * 0 * 0 0 ( 0 ) planes (0 0 ) planes ( 0 ) (0 0 ) ( 0 ) (0 0 ) * 0 ( 0 ) (0 0 ) ( 0 0) (0 0 0) a c ( 0 ) ( 0 0) a* (0 0 ) c* (0 0 0) ( 0 ) planes ( 0 ) (0 0 ) Direct Lattice ( 0 ) (0 0 ) Reciprocal Lattice 5

6 Reciprocal Space Reciprocal Lattice an a* bc sin ac sin b* c* V V cos cos cos cos * sin sin ab sin V a link between the reciprocal lattice an that connects iffraction with unit cell parameters Sphere of Reflection cos * cos * cos cos cos sin sin cos cos cos sin sin aul eter Ewal German-American ( ) place a crystal in an beam at point point can also be taken as the reciprocal lattice origin place a crystal in an beam at point point can also be taken as the reciprocal lattice origin construct a circle with a raius of / centere on the beam passing through ; is where the beam enters the circle place a crystal in an beam at point point can also be taken as the reciprocal lattice origin / / construct a circle with a raius of / centere on the beam passing through ; is where the beam enters the circle lattice point happens to be on the circle 6

7 place a crystal in an beam at point point can also be taken as the reciprocal lattice origin / / construct a circle with a raius of / centere on the beam passing through ; is where the beam enters the circle lattice point happens to be on the circle construct triangle place a crystal in an beam at point point can also be taken as the reciprocal lattice origin / / construct a circle with a raius of / centere on the beam passing through ; is where the beam enters the circle lattice point happens to be on the circle construct triangle is / hkl because it is a reciprocal lattice point is / is a right angle (triangle inscribe in semicircle) Lable as sin sin / / sin if the crystal is move to, is * hkl / hkl 7

8 sin / / sin if the crystal is move to, is * hkl / hkl is the eviation of the reflecte from the irect beam two of the family of the Miller plane with interplanar spacing are shown; makes an angle with the plane; it is the same geometry if the crystal stays at / / * hkl / hkl it is the same geometry if the crystal stays at / / by rotating the lattice (the crystal) about the origin, various reciprocal lattice points come into reflecting position; the beam shoots out in the irection that satisfies * hkl / hkl 8

it is the same geometry if the crystal stays at x h 3 by rotating the lattice (the crystal) about the origin, various reciprocal lattice points come into reflecting position; the beam shoots out in

9 it is the same geometry if the crystal stays at x h 3 by rotating the lattice (the crystal) about the origin, various reciprocal lattice points come into reflecting position; the beam shoots out in the irection that satisfies shorter allows more reflections to be coincient with sphere beam h h h 0 h h h h h 0 h h Number of possible reflections 33.6 V unit cell 3 h 3 reciprocal lattice visualize on film or etector by rotating crystal in beam an recoring s as they shoot out of sphere of reflection; reciprocal lattice parameters (a*, b*, c*, *, * an *) measure an irect lattice parameters calculate axial rotation photograph showing two imensions of reciprocal lattice 9

In the usual geometric derivation of Bragg s Law one assumes that crystalline

In the usual geometric derivation of Bragg s Law one assumes that crystalline Diffraction Principles In the usual geometric erivation of ragg s Law one assumes that crystalline arrays of atoms iffract X-rays just as the regularly etche lines of a grating iffract light. While this