Interference and Diffraction 2 2   Destructive Interference Constructive Interference

 x path difference (x)  phase difference (?) When x <  2  path difference (x)  phase difference (?) When x <  When x  

Two oscillators or two sources To point P d  dsin Phase difference when arriving at P is Original phase difference phase from path difference

What is the result of the sum? n equally spaced oscillators; equal amplitude; different in phase (different starting phase, different path length) E.g. C 2 Plane wave 1 n slits equally spaced A B a D E Huygen’s wavelets What is the result of the sum?

Using complex amplitude to solve the problem 1 1

n = 6 Q r T M O

n = 6

n = 6

1st Constructive Interference: Path difference C 2 1 A B a …. …. E D Huygen’s wavelets Phase difference Phase difference …. …. 1st Constructive Interference: 2nd Constructive Interference: In general, constructive Interference:

= I0

Width: W10 < W5 ; Height:  n2; Integrated intensity?

Fraunhofer Single Slit: d/2 x x ….. d 

} x = 0 and x = d/2; path difference BC = dsin/2. Interference between a pair of wavelets from the top and center of the slit } d/2 C B d  next pair x = 0 and x = d/2; path difference BC = dsin/2.  phase difference = (2/)*dsin/2 = dsin/ Destructive interference: dsin/ = , 2, 3, …= n.  dsin = n.

Resolution of single slit and circular aperture: The Rayleigh Criterion Well Resolved Barely Resolved http://www.kshitij-pmt.com/resolution-of-single-slit-and-circular-apertures

Single Slit: Circular aperture: Minimum  1 Airy rings

b The red ones: a The blue ones:  n sets of double slits

What about n slits each with a slit width of d? 

If d is very small, upper cap is more flat! a equals to a lot d!

Phase difference between different atoms interacted with X-ray. Diffraction Geometry 1a’, 2a’ 1 X Y 1’ 1a 2’ 2 Plane normal 3 3’   2a X’ Y’ K P M   N d L S 2 Phase difference between different atoms interacted with X-ray. Atoms in neighboring plane: 2dsin

Laue’s Equations: acos0 acos Integer Constructive interference: |acos-acos0| = h Similarly in the y direction: |bcos - bcos0| = k Similarly in the y direction: |ccos - ccos0| = l

Reciprocal lattice and diffraction: S S0 OA = pa1+qa2+ra3 p, q, r:integers S-S0 O   m n -S Path difference: uA + Av = Om+On = S0OA + (-S) OA = (S0-S)OA. v u A If b1, b2, b3: base vectors of G  constructive interference

Ewald Sphere k k k k |k| = |k| = |k| = 2/ k Diffraction codition k k k = G k k k 1/

Reciprocal lattice k k k k O