1. PHYS3321
FINDING NUCLEAR CHARGE DISTRIBUTIONS BY SCATTERING ELECTRONS – Part I
Scattering of photons from single slit
(Sections 3.3, 3.4 Dunlap)

2. Q) How would we find out the size of a slit too small to see with the eye – or even with a microscope?
A) Shine light of a fixed known wavelength on the slit and observe the diffraction pattern.

3. Single Slit DiffractionB
x.sin x dx 
-b/2
+b/2  A
Light of wavelength  is incident from the left – the positions of the wave fronts are shown. The wave is in phase along the line of the slit (x direction) . As seen from position A the in-phase light adds to give a maximum in the direction A (=0). However, at any other angle (such as B ) the light from different x-positions on the slit will be seen with different phases.-(some going up while others going down etc). To find the total amplitude () at angle  we must sum up the contributing amplitudes from all the elements “dx” taking into account their different phases .

4. Single Slit Diffraction
x.sin x dx 
-b/2
+b/2  A
“Phase angle” as seen at x
Small amplitude of wavelet coming from dx:
Integrating wave amplitude at  over the whole slit

5. Single Slit Diffraction
The Amplitude at 
The intensity at 
Where:

6. The (Sinc)2 diffraction pattern
You may think of this pattern as the “differential scattering” pattern for photons scattering from a slit.

7. Single slit differential photon scattering cross-section
By observing the position of the minima we can work out the width “b” of the slit
= = =2 =3
Minima occur when :
i.e. when:

8. From “Optics” Hect, Zajac
The same principles operate if we are dealing with photons scattering from a two dimension aperture.
The amplitude squared is the probability of finding a photon going in a specified direction.
The picture below was taken using a =633nm He-Ne laser source of photons.
The distance of the photo from the slit is 10m. The distance between minima is 0.5cm
The pattern of the diffraction pattern also reveals that the aperture is square
From “Optics” Hect, Zajac

9. Single Slit Diffraction
The Amplitude at 
Where the “Aperture function” A(x) is: 1
X=-b/2
X=+b/2
And kx = k sin() is the component of wavevector k in the x direction.
An important diffraction principle is seen: The probability of a photon going at angle  is proportional to the square of the FT of the aperture function.

10. Finding De-Broglie wavelengths
The De-Broglie wavelength associated with a particle is always given by:
(1)
For a non-relativistic particle we have K.E.
i.e.
(2)
Combining (1) and (2):
(3)
For relativistic particles (and light)
(4)
Combining (1) and (4)
(5)

11. Finding De-Broglie wavelengths
For the general case T pc E
mc2
Non-Relativistic
Relativistic