Magnetic and electrostatic deflection of charged particles 4.8.3



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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 E_o, the kinetic energy E_k and the Lorentz factor γ ≡ 1 + E_k/E_o, the above expression can be re-written as:

	BR		(γ² − 1)^1/2E_o		(1a)
			300q

or, if E_k/E_o « 1, i.e. in the non-relativistic limit,

	BR		0.144(E_km_o)^1/2		(1b)
			q

In both (la) and (lb), BR is in T·m, E_o and E_k are in MeV and m_o 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	(E_o in MeV)
	p		(γ² − 1)^1/2E_o

or, in the non-relativistic approximation,

θ(rad)		qBl		(E_k in MeV, m₀, in a.u.)
		0.144(E_km₀)^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)E_o		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, E_o in MeV)
		(γ² − 1)E_o		d

or, in the non-relativistic limit,

	θ(rad)		5 × 10⁻⁴	q	·	Vl		(V in kV, E_k in MeV)
				E_k		d

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

Particle	m₀ (a.u.)	E₀ (MeV)	\|q\|	BR (in T·m)
Particle	m₀ (a.u.)	E₀ (MeV)	\|q\|	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

G. Doucas

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