Presentation on Rutherford Scattering

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Presentation on Rutherford Scattering

• by
• Demetis Dionisis
• dionisis_at_demetis.com

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Initial Approach
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Thompsons Model

• The incident alpha particle would mostly see a
  neutral atom.
• Observations were expected to show deflections of
  less than four degrees.

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Java Applet showing the Thompsons model

• Click here to view the Applet

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The new Model (E.Rutherford)

• Geiger and Marsden (Cavendish Laboratory in
  Cambridge)
• Alpha Particles strike gold foil .
• Why did they choose gold ?

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Observation from the experiment

• Most of the particles passed through the golden
  foil with very little scattering
• Some were observed with scattering angles far in
  excess of a few degrees
• Some were backscattered !!!

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Rutherfords assumptions

• The Positive charge and almost all of the mass is
  located in a very tiny nucleus.
• The Nucleus is very dense
• The Radius of the nucleus is about 0.510(-15)
• The Nuclear radius is only 1/10000 of the atomic
  radius !

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Rutherford Scattering Java Applet

• Click here to view the applet

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I. Classical Scattering by a central potential

• We assume that the path of the particle is
  confined to the XZ plane.
• The Kinetic Energy of the particle which is being
  scattered is

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The Scattering of a particle in the field of a
repulsive central potential . The Origin of the
Potential is the Targeted nucleus.
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Important Parameters

• b Impact Parameter
• T Deflection function
• ? Angle of scattering
• a Angle between the asymptotes to the orbit

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The total Energy E Constant
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..
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The equation for the orbit ..
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The distance of the closest approach
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Having ro what is the orbit?

• Once we have obtained the distance of the closest
  approach , then we use the following integral to
  find the orbit.

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Assumptions for the acting force on the particle

• Zero at large distances
• At close distances , the particle experiences a
  repulsive or attractive force.
• ____________________________________
• As a result , the particle moves in a
  straight line trajectory (when far from the
  force) and generally gets deflected from its
  original path.

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Calculation of the angle a

• a is the angle between the two asymptotes to the
  orbit.
• or

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Deflection Function T

• T p a
• For central potentials ( Z-axis symmetry )
  , it is more convenient to use the angle of
  scattering ? ,
• ?F (if )
• 2p-F (if )

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Cross Sections

• We assume a uniform beam of particles
• Particles have the same energy and they dont
  interact with each other

• The target consists of n scattering centres
• The target particles are away from each other so
  that each scattering , implicates only one
  target.
• The target is thin enough to ignore multiple
  scattering

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dN

• n scattering centres.
• N the number of incident particles
• dN the number of incident particles that are
  scattered into the solid angle dO .
• ?hen

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What is the s(?,f) ?

• A proportionality factor in the dN equation.
• Another way to define s is
• Differential scattering cross-section
• ( ??af????? ??e???? ??at?µ? S?ed?s??)

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Total Scattering Cross-Section
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Calculation of s Step 1( out of 6 )

• We assume that the force is central and vanishes
  for large r.
• Since b is different for each particle , the
  scattering will differ too.
• Each different value of L ( or b ) defines a
  single deflection function and a scattering angle
  ?.

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Calculation of s Step 2

• The particles that are scattered between angles ?
  , ?d? are those with angular momentum between L
  , LdL or impact parameter b , bdb.
• The incident particles fall within the ring
  area 2pb db.
• The number of the particles that fall within
  the ring is N2pb db

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Calculation of s Step 3

• The number of particles that fall into the ring
  with angular momentum between L , LdL are
• And that number is equal to the number
  of the particles scattered in the dO angle.

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Calculation of s Step 4

• So we have ,

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Calculation of s Step 5
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Calculation of s Step 6

• Finally, we have two equations for s ,

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Additional Equations
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Rutherford Scattering

• Exact solution can be obtained if we use a
  Coulomb Potential .

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Repulsive Case
The closest approach distance can be
found via ,
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ro

• We pose
• and
• ..

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The Deflection function T
..........
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T
or
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The orbit of the particle
We can use the following equation ,
..
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s ?

• Since we know the connection between b and ? , we
  can calculate s via the equation ,

.
..
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Final formula for the differential cross-section