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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 (1a) 300q

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

 BR 0.144(Ekmo)1/2 (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 × 10−3 4.74 × 10−3 3.50 × 10−2 3.35 × 10−1 3.33 3.33 × 101 Proton 1.007 938.272 1 4.56 × 10−2 1.44 × 10−1 4.58 × 10−1 1.48 5.65 3.63 × 101 Deuteron 2.014 1 875.613 1 6.46 × 10−2 2.04 × 10−1 6.46 × 10−1 2.07 7.26 3.91 × 101 Alpha 4.003 3 728.4 2 4.55 × 10−2 1.44 × 10−1 4.55 × 10−1 1.45 4.85 2.20 × 101 Carbon 12.000 11 178 6 2.63 × 10−2 8.31 × 10−2 2.63 × 10−1 8.33 × 10−1 2.68 9.99

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