Diffraction by 1-D Obstacles - geometric arrangement
diffraction pattern amplitude ( )F k
all
p p ( )
( )= ( ) ik r
r
F k
∫
f re
i d rall
( ) : amplitude function
r
f r l
plane
obstacle- along -axis xz
x r ( ,0,0)
( ) ( ) x
f f
=
( ) ( ) ( ,0, )x z f r f x
k k k
→
=
r x sin scattering angle ki = k x kx=
θ
Diffraction by 1-D Obstacles
- geometric arrangement
sin
all -
( )= ( ) ik r = ( ) ikx
r
F k f r
e
d r f x e θ dx∞
∫
i∫
∞→ F(sin )
θ
= ∞∫
f x e( ) ikxsinθdx (∵k: constant)-
2
- in general, (sin ) is complexF
θ
∞
intensity of diffraction pattern (sin )F
θ
2Diffraction by 1-D Obstacles - one narrow slit
an infinite opaque sheet along the -axis containingx an infinite opaque sheet along the axis containing narrow slit at the origin
d t th l th
x
- narrowness- compared to the wavelength - amplitude function- function
δ
Diffraction by 1-D Obstacles
i i
- ( ) ( )
ik ik
f x x
θ θ
δ
∞ ∞
=
∫
sin∫
sin- -
sin
- (sin ) ( ) = ( )
1
ikx ikx
ikx
F f x e θdx x e θdx
θ
θ δ
∞ ∞
=
⎡ ⎤
∫ ∫
sin
0 2
1
- intensity ~ (sin ) 1
ikx
e x
F
θ
θ
⎡ ⎤ =
= ⎣ ⎦ =
= intensity (sin ) 1
- intensity is uniform at all angles
i l lit ti i t
F
θ
- a single narrow slit an active point source≡
Diffraction by 1-D Obstacles li
0 0
- two narrow slit
a pair of functions, one at
δ
+ x ,and the other at − x0 0
sin
- ( ) ( ) ( )
( i ) ( ) ikx
f x x x x x
f θd
δ δ
θ
∞= + + −
∫
sin-
- (sin )F
θ
f x e( ) ikx θdx∞
∞ ∞
=
∫
sin sin
0 0
- -
(x x e) ikx θdx (x x e) ikx θdx
δ δ
∞ ∞
∞ ∞
=
∫
+ +∫
−0sin 0sin 2
ikx ikx
e− θ e θ
= + = 0
( ( ) ( ) ( )
cos( sin ) )
kx
f d f
θ
∞ δ
∫
0 0-
2 2
( ( ) ( ) ( )
( i ) 4 (
) i ) F
f x x x dx f x
θ
kθ
δ
∞
− =
∫
∵
2 2
- (sin )F
θ
= 4 cos (kx0 sin )θ
Interference
Young’s Experiment
maxima sin
d θ = mλ
Diffraction by 1-D Obstacles - three narrow slits
( ) ( ) ( ) ( )
f x =
δ
x x+ 0 +δ
x +δ
x x− 0sin
( ) ( ) ( ) ( )
- (sin ) ( ) ikx
f x x x x x x
F f x e θdx
δ δ δ
θ
∞= + + +
=
∫
-
0
( ) ( )
1 2 cos( sin ) f
kx
θ
∞
= +
∫
2 2
- (sin )F
θ
= +[1 2 cos(kx0 sin )]θ
nine timesi t
more intense
Diffraction by 1-D Obstacles - narrow slits
equally spaced by a distance function N
x0 → N
δ
equally spaced by a distance function 2 1 (odd number)
x N
N p
δ
→
= +
- ( ) ( 0)
n p
n p
f x =
δ
x nx=−
=
∑
−Diffraction by 1-D Obstacles
∞
∫
sin-
sin ( 1) sin sin
- (sin ) ( ) ikx
ikpx ik p x ikpx
F f x e θdx
θ θ θ
θ
∞
=
∫
0 0 0
0 0 0
sin ( 1) sin sin
sin sin 2 sin
1
(1 )
ikpx ik p x ikpx
ikpx ikx ik px
e e e
e e e
θ θ θ
θ θ θ
− − −
−
= + + ⋅⋅⋅ + + ⋅⋅⋅ +
= + + ⋅⋅⋅ +
0 0
0 0
0
(2 1) sin s
sin sin
sin
1 1
( ) (
1
ik p x ikNx
ikpx ikpx
ikx
e e
e e
e
θ θ θ
θ
− − + − −
= =
− 0
in sin ) 1 eikx
θ
− θ
1 e
0
1 sin sin
2
e
Nkx
θ
0
2
sin sin
2
kx
θ
=
*main peak
2 0
2
sin sin
- (sin ) 2
Nkx
F
θ
θ
=0 0
angular width- 4 / separated by- 2 /
Nkx kx π π
2 0 sin
sin 2
- (sin ) F kx
θ
= θ- 2 subsidiary peaks N
Diffraction by 1-D Obstacles
0
- infinite number of narrow slits
separated by a distance x0 → infinite array of
δ
function0
p y y
- ( ) ( )
n
f x =
∑
=∞δ
x nx−- (sin ) (sin 2 )
n
n n
F k
θ δ θ π
=−∞
=
∑
=∞ − *infinitely sharp peak2π
n
∑
=−∞ kx00
(sin ) 2
kx θ π
Δ =
Grating Spectroscope
i
d sin θ λ
d θ = m λ
visible emission line of cadmium visible emission line of cadmium
visible emission line from hydrogen
Diffraction by 1-D Obstacles - one wide slit
an opaque screen containing a slit which is wide
0
~100
- ( ) 0f x if - x -X
λ
= ∞ < < 0
0 0
( ) 0 1 -
0
f x if x X
if X x X
if X
< <
< < + + 0 < < ∞
sin
0
- F(sin )= ( ) ikx
if X x
f x e θdx
θ
∞+ < < ∞
∫
0
-
F(sin ) ( )
X
f x e dx
θ
∫
∞∫
sin 0ikx X
e θ
⎡ ⎤
0
sin -
( ) ikx
X
f x e θ dx
=
∫
- 0
sin X e
ik
θ
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
Diffraction by 1-D Obstacles
0 0sin 0sin
sin
- F(sin )
sin sin
X ikX ikX
eikx e e
ik ik
θ θ
θ
θθ θ
⎡ ⎤ − −
= ⎢ ⎥ =
⎣ ⎦- 0 0
0
sin sin
sin( sin ) 2
i
ik X ik
X kX
kX
θ θ
θ θ
⎣ ⎦
= 0
0
2 2 2 0
sin
sin ( sin ) intensity~ F(sin ) 4
kX
X kX
θ
θ
= 0θ
20
0 0
- intensity~ F(sin ) 4
( sin )
- 2X (sin ) 2 /
X kX
kX
θ θ
θ π
=
→ Δ =
0 ( ) 0
- the wider slit,
the narrow the diffraction pattern the narrow the diffraction pattern - secondary peak-very rapidly weak
Single-Slit Diffraction
sin for minima sin
d m λ
δ ≈ θ θ =
http://micro.magnet.fsu.edu/primer/java/diffraction/basicdiffraction/
sin , for minima sin
d m
δ ≈ θ θ = d
Single-Slit Diffraction
- sin δ d θ
λ
≈
for minima sin m , m=1,2,3, d
θ = λ ⋅⋅⋅
Single-Slit Diffraction
http://www.phys.hawaii.edu/~teb/optics/java/slitdiffr/
Convolution
∫ ∫
- ( ) ( )* ( ) ( ) ( ) ( ) ( )
all r all r
c u = f r g r =
∫
f r g u r d r− =∫
f u r g r d r− - integrand is a function of andintegration is taken over function of
u r
r → u
g
- ( ) g r vs. (g u r− ) or ( ) g x vs. (g u x− ) for 1-D
fl ti +di l t
reflection+displacement - example
( ): function, ( ): arbitrary function
c(u)= ( ) ( - ) ( ) ( - )
f x g x
f x g u x dx x x g u x dx
δ
∞ ∞
δ
= +
∫
-∫
-0
c(u) ( ) ( ) ( ) ( )
( ) ( - )
f x g u x dx x x g u x dxo
x x g u x dx
δ δ
∞ ∞
∞
∞
+
+ −
∫ ∫
∫
-0 0
g u x( ) g u x( - )
∞
= + +
∫
inversion in 3 D
inversion in 3-D
Diffraction by 1-D Obstacles - two wide slit
0 0 0
two wide slit
slits , each of width 2 , centered at - and
( ) 0 ( )
X x x x x
f if X
= =
< < 0 + 0
0 0 0 0
- ( ) 0 - -( )
1 - ( ) -( )
f x if x x X
if x X x x X
= ∞ < < +
+ < < −
0 0 0 0
0 0
0 - ( ) ( ) 1 ( )
if x X x x X
if x X x
− < < −
− < < (x0 + X0)
0 0
0 ( )
- ( ) is convolution of one wide slit and two narrow slit
if x X x
f x
+ < < ∞
( ) is convolution of one wide slit and two narrow slit
( ) ( )* ( )
f x
f two wide slit = f one wide slit f two narrow slit
Diffraction by 1-D Obstacles
- Fourier transform
( ) [ ( )* ( )]
- Fourier transform of a convolution is the product of
Tf two wide slit = T f one wide slit f two narrow slit the individual Fourier transforms
( ) (
Tf two wide slit = Tf one wide slit Tf two narrow slit)⋅ ( )]
(Tf two wide slit) Tf one wide( ) ( )]
- one wide slit
sin( sin )
slit Tf two narrow slit
kX0
θ
0
0
sin( sin ) (sin ) 2
sin
F X kX
kX
θ θ
=
θ
0
- two narrow slit
(sin ) 2cos(F
θ
= kx sin )θ
Diffraction by 1-D Obstacles id li
0
0 0
- two wide slit
sin( sin )
(sin ) 4 kX cos( sin )
F
θ
= X0θ
kx0θ
0
(sin ) 4 cos( sin )
sin -intensity
F X kx
θ
kXθ
θ
2 2 2 0 2
0 2 0
0
sin ( sin )
(sin ) 16 cos ( sin )
( sin )
F X kX kx
kX
θ θ θ
=
θ
(kX0 sin )
θ
Diffraction by 1-D Obstacles
- two wide slit
2 2
form of diffraction pattern is a series of cos fringes modulated by (sin / )
α α
2
0 1
y ( )
- cos function- first zero sin / 2
sin /(2 )
kx kx
θ π θ π
=
1 = 0
2
0 2
sin /(2 )
- (sin / ) function- first zero sin kx
kX
θ π
α α θ
=
=
π
2 0
0 0 2 1
sin /( )
- x
kX X
θ π
θ θ
=
→ >
0 0 2 1
Diffraction by 1-D Obstacles - three wide slit
- amplitude function
( ) ( )* ( )
( ) [ ( )* ( )]
f three wide slit f one wide slit f three narrow slits
Tf threeo wide slit T f one wide slit f three narrow slits
=
=
( ) [ ( ) ( )]
( ) ( )* (
Tf threeo wide slit T f one wide slit f three narrow slits Tf threeo wide slit = Tf one wide slit Tf )
sin( sin )
three narrow slits kX0
θ
0 0
0
sin( sin )
- (sin ) 2 [1 2cos( sin )]
sin
F X kX kx
kX
θ θ θ
=
θ
+Diffraction by 1-D Obstacles - three wide slits
- intensity
2 2 2 0 2
0 2 0
y
sin ( sin )
(sin ) 4 [1 2cos( sin )]
( sin )
F X kX kx
kX
θ θ θ
=
θ
+(kX0 sin )
θ
Diffraction by 1-D Obstacles
0
- wide slits
width of 2X , centered on a function N
0,
δ
- amplitude function
( ) ( )* ( )
f N wide slit = f one wide slit f N narrow slits
( ) ( )* ( )
( ) [ ( )* ( )]
( )
f N wide slit f one wide slit f N narrow slits Tf N wide slit T f one wide slit f N narrow slits
Tf N d l T
=
=
( )* ( )
f d l Tf N l
( Tf N wide slit) = Tf one wide slit Tf N narrow slits( )* ( )
Diffraction by 1-D Obstacles - N wide slits
- intensity
2 0
2 2 2 0
sin sin
sin ( sin ) 2
(sin ) 4
Nkx
F X kX
θ
θθ
=2 0
0 2
0 sin
sin 2
(sin ) 4
( sin ) kx
F X
kX θ
θ
=θ
Diffraction by 1-D Obstacles - infinite wide slits
li d f i - amplitude function
( f ∞ wide slit) = f one wide slit( )* ( f ∞ narrow slits)
( ) [ ( )* ( )]
( ) ( )* (
Tf wide slit T f one wide slit f narrow slits Tf wide slit Tf one wide slit Tf narrow slit
∞ = ∞
∞ = ∞ s)
( ) ( ) (
f f f )
Diffraction by 1-D Obstacles
- infinite number of wide slits
2 2
2 2 0
- intensity
sin ( sin ) 2
( i
θ
) 2 4 02 kX0θ
2 n∑
=∞δ
( iθ
nπ
)0 0
sin ( sin ) 2
(sin ) 4 (sin )
( sin ) n
kX n
F X
kX kx
θ π
θ δ θ
θ
=−∞=
∑
−Diffraction by 1-D Obstacles - significance of the diffraction pattern
Diffraction by 1-D Obstacles - narrow slits
1 As the number of slits increases the main peaks become 1. As the number of slits increases, the main peaks become sharper and narrower, whilest the subsidiary peaks
become rapidly less intense
2. The position and separation of the main peaks is constant independent of the number of peaks. The man peaks are separated by an angular deflection given byp y g g y
(sin ) 2
kx
θ π
Δ =
0
The distance is determined soly by and separation of0
kx
λ
xi hb i li
any two neighboring narrow slits.
Diffraction by 1-D Obstacles
- narrow slits
- The position of the main peak in a diffraction pattern is determined solely by the lattice spacing in an obstacle.y y p g - The shape of the main peak is determined by the overall
shape of the obstacle shape of the obstacle.
Diffraction by 1-D Obstacles - wide slits
- The effect of the motif (one wide slit) is to alter the The effect of the motif (one wide slit) is to alter the intensity of each main peak, but the position of the
i k h d
main peaks are unchanged - intensity envolope
→ structure of motif
Diffraction by 1-D Obstacles - another way of looking at N wide slits
h f i
- shape function
is zero everywhere outside an obstacle
corresponds to the macroscopic shape of the obstacle within the obstacle
Diffraction by 1-D Obstacles - finite lattice
( ) ( ) ( )
f finite lattice = f infinite lattice f shape functioni
( ) ( ) ( )
( ) ( )* ( )
f finite lattice f infinite lattice f shape function f obstacle f motif f finite lattice
=
=
i
( ) ( )*[ ( ) ( )]
- diffraction pattern
f obstacle = f motif f infinite lattice f shape functioni
(sin ) ( )
{ ( )*[ ( ) ( )]}
F Tf obstacle
T f motif f infinite lattice f shape function
θ
== { ( ) [ ( ) (i )]}
( ) [ ( ) ( )]
( ) [ ( )* ( )]
f f f f f p f
Tf motif T f infinite lattice f shape function Tf motif Tf infinite lattice Tf shape function
=
=
i i
i
= Tf motif( ) [ (i Tf infinite lattice Tf shape function) ( )]
Diffraction by 1-D Obstacles - N wide slits
( ) ( )* ( )
f N wide slits = f one wide slit f N narrow slits
( ) ( ) ( )
( ) ( )*
[ ( ) ( )]
f N wide slits f one wide slit f N narrow slits f N wide slits f one wide slit
f lit f h f ti
=
=
[ ( f ∞ narrow slits f shape function) (i )]
Diffraction by 1-D Obstacles - diffraction pattern
( ) { ( )*
Tf N wide slits T f one wide slit
( ) { ( )*
[ ( ) ( )]}
Tf N wide slits T f one wide slit
f narrow slits f shape function
=
∞ i
= Tf one wide slit T f( ) [ ( i ∞ narrow slits f shape function) (i )]
= Tf one wide slit Tf( )i ( ∞ narrow slits Tf shape function)* ( ) - three types of structural informationyp
that concerning the lattice that concerning the motif
that concerning the motif
that concerning the shape of the entire crystal
Diffraction by 1-D Obstacles
Diffraction by 1-D Obstacles
Summary
the position of the main peaks gives information - the of the main peaks gives information on the
position lattice
- the shape of each main peak gives information on the overall object shape
set of in
- the tensities of all the main peaks gives information on the structure of the motif
on the structure of the motif
