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Chapter: 4 Atomic and nuclear physics
    Section: 4.8 Nuclei and particles
        SubSection: 4.8.2 Rutherford scattering

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4.8.2 Rutherford scattering

Rutherford scattering occurs when a high energy (MeV) ion is deflected through a large angle by the repulsive electrostatic field of an atomic nucleus. True Rutherford scattering occurs outside the nucleus. If the incoming ion has sufficient kinetic energy to penetrate the nucleus, the scattering probability is modified by a nuclear interaction. An approximate empirical criterion for avoiding nuclear interaction is given by:

spcaer

 E0 < Z1Z2/(A11/3 + A21/3)   

    (1)

where E0 is the kinetic energy of the incoming ion (in MeV), Z1, and Z2 are the atomic numbers and A1 and A2 are the nucleon numbers of the incoming ion and the target atom, respectively. It is assumed here that the incoming ion energy is low enough that true Rutherford scattering occurs outside the nucleus, but high enough to avoid significant effects of screening by electrons, which, at large distances of approach, reduce the effective charge of the nucleus. These criteria are suitably met, for example, by He+ ions with an energy of around 1 MeV.



The energy E1 of a back-scattered ion immediately after the collision is given by:

spcaer

E1 = kE0   

    (2)

where

spcaer

k =

(M22M12 sin2 θ)1/2 + M1 cos θ

2

    (2a)

M1 + M2

is called the kinematic factor. M1 and M2 are the relative atomic masses of the incoming ion and the target atom, respectively, and θ is the angle of the scattered ion with respect to the incoming beam direction. For a scattering angle close to 180°, equation (2) approximates to:

spcaer
E1 = E0

M1M2

2

  

M1 + M2

    (2b)


There is a one-to-one correlation between the scattering angle, θ, and the perpendicular distance, b, known as the impact parameter, between the incoming path of the ion and a parallel line passing through the nucleus. The distance of closest approach of the ion to the scattering nucleus occurs for 180° scattering and is smaller for higher energy ions and from nuclei lower in atomic number.

The probability of scattering of a positive ion by a positive nucleus is described by the Rutherford scattering cross section:

spcaer σ (θ)=

Z1Z2e2

2

 

4

  (a + cos θ)2

      (3)

4E0

sin4θ

a

where

spcaer a = [1 − ((M1/M2) sin θ)2]1/2   

and e is the unit electric charge. The probability of scattering is proportional to the square of the atomic number of the target atom and inversely proportional to the square of the ion energy. Therefore scattering has higher probability for low energy ions and heavy target atoms.

The tables show (1) values of backscatter energy for three different ions at an incident energy of 1 MeV and (2) values of the Rutherford cross-section for 1 MeV 4He ions scattered from gold as a function of laboratory scattering angle.

Table 1. Energies (in MeV) of ions backscattered by 180° from various target atoms for an incident ion energy of 1 MeV
Note that the scattering cross-section for 12C is likely to be affected by screening effects at this energy.

    Target element

12C

16O

28Si

40Ca

56Fe

107Ag

181Ta

197Au

238U

 

 

 

 

 

 

 

 

 

 

Ion

 

 

 

 

 

 

 

 

 

     1H +      .    .    .    .    .    .    .    .    .

0.714

0.777

0.866

0.904

0.930

0.963

0.978

0.980

0.983

     4He +   .    .    .    .    .    .    .    .    .

0.250

0.360

0.562

0.669

0.751

0.861

0.915

0.922

0.935

     12C +   .    .    .    .    .    .    .    .    .

0.020

0.160

0.290

0.418

0.637

0.767

0.783

0.817

 

                 

Table 2. Variation of Rutherford cross-section σ(θ) in units of 10−27 m2 sr−1 and scattered energy E1 in MeV for 1 MeV 4He ions scattering from gold, as a function of scattering angle θ in the laboratory


    θ

20°

40°

60°

90°

120°

150°

170°

175°

 

 

 

 

 

 

 

 

 

σ(θ)    .    .    .    .    .

3558       

236.4    

51.76    

12.94    

5.749

3.714

3.282

3.245

E1      .    .    .    .    .

      0.998

    0.991

0.980

0.960

0.941

0.927

0.922

0.922

 

               


Reference

W.-K. Chu, J. W. Mayer and M.-A. Nicolet (1978) Backscattering Spectrometry, Academic Press, New York.

J.Asher

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