Particle equations

2.2 General envelope equation of charged particle beam

2.2.1 Particle equations

They begin with the equations of motion of a single particle (rest massm, chargeq) traveling through a system of lenses, gas cells, foils, etc. The axis of rotational symmetry is taken aszaxis of cylindrical coordinates (r,φ,z). Particle momenta are mostly in the propagation directionz. The motion is governed by the Lorentz force law with an added term representing the scattering field of the medium:

˙p=q(E+v×B) +δF. (2.21) HereδFis the rapidly fluctuating force due to the individual particles of the gas. The dot notation is the total time differentiation. The d/dt notation is used for avoiding ambiguity in certain situations. The relativistic momentum is

= =γm˙r,

γ= (1−β²)^−1/2, β =|v|

c . (2.23)

The symbolsEandBindicate the macroscopic fields generated by external coils, charges, and currents of beam and those induced from gas. From Eqs. (2.21) to (2.23) we obtain the equation for the rate of relativistic energy changeW˙ ≡γmc²:

W˙ =qv·E+v·δF. (2.24)

Proof 1

˙p=γm˙v+γmv˙ = (1+γ²β²)γm˙v=γ³m˙v, W˙ =γmc˙ ²=v·γ³m˙v=v·˙p=qv·E+v·δF.

The paraxial approximation simplifies the equations of motion. The velocity is decomposed into longi- tudinal and transverse components,

v=v_zzˆ+v⊥ (2.25)

with the assumptionv_⊥≪v_z. For a self-pinched beam, this approach requires the beam currentI_bto be small compared with the Alfvén limit,

IA=β γmc4π

qµ₀ =β γqc

r₀ ≈17β γ kA,

wherer₀is the classical radius of the electron. We characterize the motion with the small parameter¹∆:

∆²= I_b IA

≪1, v_⊥

v_z =O(∆).

The depth of the electrostatic potential well of the beam divided by the particle kinetic energy, is also on the order of∆².

Proof 2

The radial equation of motion for an electron with kinetic energy(γ₀−1)mec²and zero angular momen- tum is²

dVR

dτ =− I

[4π ε0mec³βAγA/e]

2VZF(R) R with

I_A≡4π ε₀m_ec³βAγA

e .

1It is also assumed thatBzis not large compared withB_θ/∆. There are special cases in which the paraxial approximation applies even through∆≳1, for example, a counter-flowing plasma current may cancel some of the beam current.

2See Stanley Humphries, Jr.,Charged Particle Beams(2002)

DefiningI=eNβcandre=e²/4π ε0mec², Nre

γ = e²N 4π ε₀m_ec²γ =

I I_A

γAβA

γ β

,

whereγAβA/γ β is close to unity far from the Alfven current limit. The difference in electrostatic poten- tial between the axis and the beam envelope of the solid beam is

△φmax≈en₀ 4ε₀

r²₀.

Then the thickness of the electrostatic potential well of the beam space charge divided by the beam particle kinetic energy is

△φmax

γm_ec² ≈ e²N

4π ε₀m_ec²γ =Nre

γ ≡∆² with N=n₀r²₀π.

In order to calculate the order ofv_⊥/vz, the fraction of the net current of the simplified uniform beam profile can be written as below.

F(R) =F r

r₀

= N N₀ ≈

r r₀

2

=R². So, our equation of motion is specified to

d²R dτ² =−I

IA

2VZF(R)

R ≈ −

Nr_e γ 2VZ

R.

AssumingV_Zis constant with the time scale of transverse dynamics, R(τ) =R₀cos(Dτ+α) with D²=Nr_e

γ 2VZ, and

VR(τ) =−DR₀sin(Dτ+α).

Finally the order ofv⊥/vzis v⊥

v_z =VR

V_Z ∝ D V_Z =

r 2 V_Z

s Nre

γ ∝ s

Nre

γ =∆.

HereVZ≈1 was used.

From Eq. (2.23) we have

v²_z=c²

1− 1 γ²

−v²_⊥=β²c²+O(∆²). (2.26) Similarly, the energy equation (2.24) becomes

W˙ =q(vzEz+v⊥·E⊥) +v·δF=qvzEz+δW

δt +O(∆²), (2.27)

whereδW/δtis the known drag due to ionization and excitation of the medium. A segment of beam at givenzandtis characterized by a single value ofW (equivalentlyγ), so Eq. (2.27) is averaged over the current profile of the segment to get

γ˙= qvz

mc²E¯z+ 1 mc²

δW

δt . (2.28)

Variablesvzandγare thus regarded as known functions of z and t, independent of the transverse motion.

The dynamics is simplified, but there is no treatment of the electrostatic lens. The transverse components of the equations of motion (2.21) and (2.22) are

˙ γ

γv⊥+v˙⊥= q

γm(E⊥+βcˆz×B⊥+v⊥×Bzz) +ˆ 1

γmδF⊥ (2.29)

and

˙r_⊥=v⊥. (2.30)

It is advantageous to use the cylindrically resolved field components in Eq. (2.29). The components are observed as below:

1. Bzis a known function of z and t, generated by coils surrounding the region of propagation. Since B_zis assumed not to depend on r, the forceqv⊥×B_zzˆis linear. This property always holds near the axis and no significant error occurs in extending it over the beam profile if the scale length of change is large compared with the beam radius. The beam itself may be a source ofB_z, but the contribution isO(ε²)compared withB_θ and may be neglected.

2. Bris determined from the Maxwell equation∇·B=0. Recalling that by symmetry∂B_θ/∂ θ=0, we have

Br=−r 2

∂B_z

∂z . (2.31)

Proof 3

∇·B=1 r

∂

∂r(rBr) +1 r

∂B_θ

∂ θ +∂Bz

∂z =0.

Considering the symmetry, the second term goes to zero.

∂

∂r(rBr) =−r∂Bz

∂z .

From the paraxial approximation, assuming∂Bz/∂zisrindependent, rBr=−r²

2

∂Bz

∂z .

3. E_θ is similarly determined (from Faraday’s law) to be E_θ =−r

2

∂Bz

∂t . (2.32)

Proof 4

∇×E= 1

r

∂Ez

∂ θ −∂E_θ

∂z

ˆ r+

∂Er

∂z −∂Ez

∂r

θˆ+1 r

∂

∂r(rEθ)−∂Er

∂ θ

ˆ

z=−∂B

∂t.

Considering the symmetry and the orders of the components, only the first term of the third bracket in the left-hand side survives.

1 r

∂

∂r(rE_θ) =−∂Bz

∂t . As above, the final result is derived.

4. Er and B_θ are generated by the charges and currents of the beam and background gas. These components need not be specified in detail at this point.

5. The field E_z is generated by the beam, gas, and external sources and, in general, is O(cB_θ).

However, it acts directly to changeγandvzrather thanv_⊥, so its effect is small and on the order of∆². A cumulative energy change results from the time average E_z during propagation over a long distance. The effect has been taken into account in Eq. (2.28). The use of the spatial as well as time average⟨E_z⟩in Eq. (2.28) is justified by the fact that, due to betatron oscillations, a particle is not localized at a single value ofrduring propagation.

In terms of the above field components, the transverse equation of motion is γ˙

γv⊥+˙v⊥= q γm

E_rrˆ−r

2

∂Bz

∂t θˆ−βcr 2

∂Bz

∂z zˆ×rˆ+βcB_θzˆ×θˆ−B_zzˆ×v⊥

+ 1

γmδF⊥. (2.33) Noting that

∂Bz

∂t +βc∂Bz

∂z =dBz

dt =B˙z, (2.34)

(where ˙Bz is the convective derivative, i.e., the variation ofBz experienced by a moving particle), we may write

˙ γ

γv_⊥+˙v_⊥= q γm

(Er−βcB_θ)rˆ−B˙z

2 zˆ×r_⊥−Bzzˆ×v_⊥

+δF⊥

γm . (2.35)

At this point we define the relativistic cyclotron and betatron frequencies, ωc=qBz

γm and ω_β²= q γm

βcB_θ−Er

r . (2.36)

In generalω_β²is a function ofr, although for a flat current and charge profile it is a constant. In a vacuum, ω_β²is negative. Employing these definitions, Eq. (2.35) may be written

˙ γ

γv+˙v+ω_β²r+ωczˆ×v+ 1 2γ

d

dt(γ ωc)zˆ×r=δF

γm. (2.37)

Here and henceforth we drop the⊥notation for convenience.