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Chapter: 4 Atomic and nuclear physics
    Section: 4.8 Nuclei and particles
        SubSection: 4.8.3 Magnetic and electrostatic deflection of charged particles

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4.8.3 Magnetic and electrostatic deflection of charged particles

Bending radii in magnetic fields

A particle of rest mass m0, momentum p and charge qe, where e is the electron charge and q an integer, that enters a uniform magnetic field, B, perpendicular to its velocity, will describe a circular orbit of radius, R; its ‘magnetic rigidity’, BR, is given by the expression:

BR

p

qe


Using the particle's rest energy Eo, the kinetic energy Ek and the Lorentz factor γ ≡ 1 + Ek/Eo, the above expression can be re-written as:

   

BR

  

(γ2 − 1)1/2Eo

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(1a)

300q

or, if Ek/Eo « 1, i.e. in the non-relativistic limit,

    

BR

0.144(Ekmo)1/2

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(1b)

q

In both (la) and (lb), BR is in T·m, Eo and Ek are in MeV and mo is in atomic mass units. See table below for some typical values.

Magnetic deflectors
In the small deflection angle θ limit, tan θ θ, a magnetic deflector of length l in metres, and field strength B, in Tesla, will produce a deflection

θ(rad) = 

qeBl

  

300qBl

  (Eo in MeV)

p

(γ2 − 1)1/2Eo

or, in the non-relativistic approximation,

θ(rad)

 

qBl

   

(Ek in MeV, m0, in a.u.)

0.144(Ekm0)1/2


Electrostatic deflectors

For a particle of velocity v, perpendicular to the field of a parallel-plate deflector of length l, gap d and deflecting potential V, the deflection angle θ is given by:

 θ  =

qe

 ·

Vl

  = 

qeγ

 ·

  Vl

pv

d

(γ2 − 1)Eo

d

where the other symbols are defined as before; it is assumed that tan θ θ.

For practical applications:

 θ(rad) = 

10−3

qγ

 ·

  Vl

    

(V in kV, Eo in MeV)

(γ2 − 1)Eo

d

or, in the non-relativistic limit,

   

θ(rad)

5 × 10−4

q

 ·

  Vl

    

(V in kV, Ek in MeV)

Ek

  d




Table of magnetic rigidities, BR (in T·m) for different values of Ek (in MeV)

Particle 

m0 
(a.u.)

E0 
(MeV)

  |q| 

BR (in T·m)

0.1 Mev

1 Mev

10 Mev

100 Mev

1 GeV

10 GeV

 

 

 

 

 

 

 

 

 

 

Electron

5.486 × 10−4

       0.511

  1

1.12 × 103

4.74 × 103

3.50 × 102

3.35 × 10−1

3.33

3.33 × 101

Proton

 1.007

   938.272

  1

4.56 × 102

1.44 × 101

4.58 × 101

 1.48

5.65

3.63 × 101

Deuteron

 2.014

1 875.613

  1

6.46 × 102

2.04 × 101

6.46 × 101

 2.07

7.26

3.91 × 101

Alpha

 4.003

3 728.4    

  2

4.55 × 102

1.44 × 101

4.55 × 101

 1.45

4.85

2.20 × 101

Carbon

12.000 

11 178         

  6

2.63 × 102

8.31 × 102

2.63 × 101

 8.33 × 101

2.68

9.99

 

 

 

 

 

 

 

 

 

 

G. Doucas

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