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Interference and Diffraction. 2 . 2 . 0. 0. . . Destructive Interference. Constructive Interference. x. 2 . 0. . 0. path difference ( x )  phase difference (?). When x < . When x  . Two oscillators or two sources. To point P. . d. d sin .

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Interference and Diffraction

2

2

0

0

Destructive Interference

Constructive Interference


x

2

0

0

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


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

E

D

Huygen’s

wavelets

What is the result of the sum?



n = 6

Q

r

T

M

O


n = 6


n = 6


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:

0

d/2

x

x

…..

d


Interference between

a pair of wavelets

from the top and

center of the slit

}

0

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?

d

a


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


Diffraction Geometry

1a’, 2a’

X

Y

1

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:

a

0

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

b1, b2, b3: base vectors of G

If

 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


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