Question 13.11.Q1: Several conditions must be met for particle acceleration wit......

(a) Three general conditions for particle acceleration in any type of particle accelerator are:

(1) Particle to be accelerated must be charged. All particles generated by accelerators are accelerated by electric fields, be it electrostatic fields in electrostatic accelerators or electromagnetic fields in cyclic accelerators. Particles with positive charge move in the direction of electric field, negatively charged particles move in direction opposite to the electric field. Neutral particles cannot be accelerated with electric fields.

(2) Electric field used for particle acceleration must be oriented in the direction of propagation of the charged particle for positively charged particles and in direction opposite to propagation for negatively charged particles.. In a particle accelerator the trajectory of the accelerated particle is: (i) linear, (ii) circular, or (iii) in the shape of racetrack.

(i) Linear trajectory is found in x-ray tube, neutron generator, Van de Graaff
generator, and linear accelerator.
(ii) Circular trajectory is found in betatron, cyclotron, and microtron.
(iii) Racetrack trajectory is used in microtron and synchrotron.

(3) Particle must be accelerated in vacuum rather than in a dielectric material to avoid deleterious collisions between the accelerated particle and atoms of the medium in which the accelerated particle is traveling.

(b) Three conditions that must be met for acceleration of electrons in an acceleration electromagnetic (EM) waveguide of a linear accelerator (linac).

(1) The radiofrequency (RF) mode to be used for electron acceleration in an EM waveguide must provide a finite, non-zero value electric field component {E}_{z0} at r = 0 to enable the electron acceleration along the central axis of the waveguide. Of the three special modes of relevance to RF propagation:

(i) Transverse electric \left(TE_{mn}\right) mode for which {E}_z = 0 everywhere in waveguide core,
(ii) Transverse magnetic \left(TM_{mn}\right) mode for which \mathcal{B}_z = 0 everywhere in the waveguide core,
(iii) Transverse electromagnetic \left(TEM_{mn}\right) mode for which {E}_z = \mathcal{B}_z = 0,

only two modes (TE and TM) can propagate in an EM waveguide, and only one (TM) of these two modes produces non-zero electric field along the central axis of the waveguide, i.e., {E}_z|_{r=a} \neq 0. Thus, TM_{mn} modes are used for electron acceleration and, of all possible m and n values, the lowest ones are used for electron acceleration (m = 0 and n = 1). Of all possible angular frequencies ω, the angular frequency used for acceleration must be in the first Brillouin zone between cutoff frequencies ω_{c1}\ and\ ω_{c2} and of such magnitude that produces a phase velocity υ_{ph} less than and approximately equal to speed of light c in vacuum and a group velocity υ_{gr} larger than zero.
While TM_{mn} modes can propagate in a uniform EM waveguide, the phase velocity υ_{ph} in a uniform waveguide exceeds c making uniform waveguides unsuitable for charged particle acceleration. To achieve υ_{ph} ≤ c one uses a uniform waveguide modified with periodic perturbations (disks or irises) that effectively decrease υ_{ph} to a level below c and this type of waveguide is referred to as disk-loaded or acceleration waveguide.

(2) To allow the RF wave to accelerate electrons the phase velocity υ_{ph} of the RF wave must be equal to the particle velocity υ_{part}. Thus, the electron generated by the electron gun must be injected into the acceleration waveguide with a certain velocity υ_{part} [corresponding to a certain kinetic energy \left(E_K\right)_{inj}] that matches the phase velocity of the RF wave. Since: (i) the accelerated particle velocity cannot exceed c and (ii) υ_{part} must be matched with υ_{ph}, it becomes obvious that υ_{ph} of the RF wave used for particle acceleration in an acceleration waveguide should be equal or less than c, i.e., υ_{ph} ≤ c.

(3) Condition (2) states that for electron acceleration with RF fields υ_{part} ≈ υ_{ph}; however, the electron is injected into the acceleration waveguide with a relatively low velocity υ_{part} [that is with relatively low kinetic energy \left(E_K\right)_{inj}] that is substantially smaller than c and is then accelerated in the waveguide to final relativistic kinetic energy corresponding to final velocity ∼c. Thus, the condition υ_{part} ≈ υ_{ph} cannot be fulfilled easily at the entrance side of the waveguide.
There are two possible solutions to this problem: one is to lower the phase velocity of the RF wave υ_{ph} on the electron gun side of the acceleration waveguide to obtain υ_{inj} ≈ υ_{ph} and then gradually increase the phase velocity υ_{ph} toward c as the accelerated charged particle gains kinetic energy. This approach is referred to as velocity modulation of the RF wave.
The other solution is to provide sufficiently large amplitude of the electric field {E}_{z0} for the wave to capture the electron at the entrance to the accelerating waveguide despite its relatively low injection velocity υ_0 that is significantly smaller than the phase velocity υ_{ph} of the RF wave.
Of the two options, the first one is more difficult as it involves modulation of the phase velocity υ_{ph} by using non-uniform cavities in the entrance section of the acceleration waveguide and uniform cavities farther down the waveguide. Early linac designs contained many cavities with varying inner diameter, aperture radius, and axial spacing; more recently, only a few cavities were used for this purpose, and currently, a single half-cavity provides the phase modulation. The improved understanding of velocity modulation has resulted in a substantial lowering of the required gun injection voltage from historical levels of above 100 kV to current levels of around 25 kV.
The second approach to υ_0 < υ_{ph} ≈ 0 is based on the calculation of the minimum amplitude of the electric field [({E}_z)_0]_{min} that still allows the RF wave to capture the electron injected with a relatively low velocity υ_{inj} from the electron gun into the acceleration waveguide. The larger is [(E_z)_0]_{min} in the waveguide the lower is the required injection velocity or injection kinetic energy of the electron entering the waveguide. The relationship between [(E_z)_0]_{min} and υ_0 is referred to as the capture condition for an acceleration EM waveguide and must be satisfied for the acceleration to proceed.

(c) The capture condition is derived from the relativistic equation of motion of the electron in the electric field with the help of two simplifying assumptions:

(1) RF wave propagates through the acceleration waveguide with a phase velocity υ_{ph} approximately equal to c (i.e., υ_{ph} ≈ c).

(2) Electric field {E}_z is in the direction of propagation and has a sinusoidal behavior in time, such that

{E}_z=\left({E}_z\right)_0 \sin \varphi,             (13.302)

with {E}_{z0} the amplitude of the electric field and ϕ the phase angle between the wave and the electron, given as:

\varphi=k_{\mathrm{g}} z-\omega t,          (13.303)

where

ω is the angular frequency of the wave.
k_g is the waveguide wave number or propagation coefficient.
z is the coordinate along the waveguide axis.

The rate of change d/dt of phase ϕ with time t is from (BB) given as

\frac{\mathrm{d} \varphi}{\mathrm{d} t}=k_{\mathrm{g}} \frac{\mathrm{d} z}{\mathrm{~d} t}-\omega=k_{\mathrm{g}} v_{\text {part }}-\omega=\frac{2 \pi}{\lambda_{\mathrm{g}}}(\beta-1)          (13.304)

with υ_{ph} ≈ c [assumption (1) above], k_g = 2π/λ_g\ where\ λ_g is the RF wavelength, and β = υ_{part}/c is the electron velocity υ_{part} normalized to speed of light c in vacuum. The relativistic equation of motion for the electron moving in the electric field {E}_z may be written as

F=\frac{\mathrm{d} p}{\mathrm{~d} t}=\frac{\mathrm{d}}{\mathrm{d} t} m(v) v=\frac{\mathrm{d}}{\mathrm{d} t} \frac{m_{\mathrm{e}} \beta c}{\left(1-\beta^2\right)^{1 / 2}}=e {E}=e\left({E}_z\right)_0 \sin \varphi,        (13.305)

where

F force exerted on the electron by the electric field {E}_z.
p electron momentum.
m(υ) mass of the electron at velocity υ.
m_e electron rest mass (0.511 MeV).

Equations (13.304) and (13.305) are now simplified as follows

\frac{\mathrm{d} \varphi}{\mathrm{d} t}=a(\beta-1)             (13.306)

and

\frac{\mathrm{d}}{\mathrm{d} t} \frac{\beta}{\left(1-\beta^2\right)^{1 / 2}}=b \sin \varphi         (13.307)

respectively, introducing parameters a and b defined as a = 2πc/λ_g\ and\ b = e\left({E}_z\right)_0/\left(m_{e}c\right).
Introducing a new variable cosα = β into (13.306) and (13.307) we get, respectively,

\frac{\mathrm{d} \varphi}{\mathrm{d} t} \equiv \frac{\mathrm{d} \varphi}{\mathrm{d} \alpha} \frac{\mathrm{d} \alpha}{\mathrm{d} t}=a(\cos \alpha-1) \quad \text { or } \quad \frac{\mathrm{d} \alpha}{\mathrm{d} t}=a(\cos \alpha-1) \frac{\mathrm{d} \alpha}{\mathrm{d} \varphi}        (13.308)

and

\frac{\mathrm{d}}{\mathrm{d} t} \frac{\cos \alpha}{\sin \alpha} \equiv \frac{\mathrm{d}}{\mathrm{d} t} \cot \alpha \equiv \frac{\mathrm{d} \cot \alpha}{\mathrm{d} \alpha} \frac{\mathrm{d} \alpha}{\mathrm{d} t} \equiv \frac{1}{\sin ^2 \alpha} \frac{\mathrm{d} \alpha}{\mathrm{d} t}=b \sin \alpha \quad \text { or } \quad \frac{\mathrm{d} \alpha}{\mathrm{d} t}=-b \sin ^2 \alpha \sin \varphi \text {. }        (13.309)

After equating the two expressions above for dα/dt, rearranging terms, and integrating over ϕ from initial ϕ_0 to ϕ and over α from initial α_0 to α, we get

that results in

After inserting cosα = β and \cos α_0 = β_0, and recognizing that at the end of the acceleration process β ≈ 1, we obtain

\cos \varphi-\cos \varphi_0=\frac{a}{b}\left(\frac{1-\beta_0}{1+\beta_0}\right)^{1 / 2}=\frac{2 \pi}{\lambda_{\mathrm{g}}} \frac{m_{\mathrm{e}} c^2}{e\left(E_z\right)_0}\left(\frac{1-\beta_0}{1+\beta_0}\right)^{1 / 2},        (13.312)

where β_0 = υ_0/c with υ_0 the initial (injection) velocity of the electron injected into the accelerating waveguide from the electron gun. Since the left-hand side of (13.312) cannot exceed 2, we obtain the following relationship for the capture condition

\left({E}_z\right)_0 \geq \frac{\pi m_{\mathrm{e}} c^2}{\lambda_{\mathrm{g}} e} \sqrt{\frac{1-\beta_0}{1+\beta_0}}=\frac{K}{\lambda_{\mathrm{g}}} \sqrt{\frac{1-\beta_0}{1+\beta_0}}         (13.313)

where K = πm_ec^2/e = 1.605 MV is the capture constant of the electron in a disk-loaded acceleration waveguide. The minimum amplitude of the electric field [({E}_z)_0]_{min} is thus expressed as follows

\left[\left({E}_z\right)_0\right]_{\min }=\frac{K}{\lambda_{\mathrm{g}}} \sqrt{\frac{1-\beta_0}{1+\beta_0}} .           (13.314)

Equation (13.314) is referred to as the capture condition and must be satisfied, if an electron entering the acceleration waveguide from the electron gun with initial velocity υ_0 is to be captured by the radiofrequency wave that has a phase velocity close to c.
The well known relativistic relationship between the electron initial (injection) velocity β_0 and the electron initial kinetic energy \left(E_K\right)_0 is given as follows [see (T2.7)]

\beta_0=\sqrt{1-\frac{1}{\left(1+\frac{\left(E_{\mathrm{K}}\right)_0}{m_{\mathrm{e}} c^2}\right)^2}}           (13.315)

allowing us to estimate [({E}_z)_0]_{min}, the minimum amplitude of the radiofrequency field, for typical gun injection voltage potentials in the range from 20 keV to 100 keV.

(d) The capture condition (13.314), relating (i) minimum amplitude [({E}_z)_0]_{min} of electric field required to capture an electron injected into an acceleration EM waveguide and (ii) velocity β_0 of an electron injected into the waveguide from the electron gun, is given in (13.293). The range covered by the capture condition extends from 0 to 1 for normalized velocity β and the limiting values of [({E}_z)_0]_{min} are given as follows:

(1) The upper limit of [({E}_z)_0]_{min} is attained when β_0 → 0 corresponding to υ_0 = 0

\lim _{\beta_0 \rightarrow 0}\left[\left({E}_z\right)_0\right]_{\min }=\lim _{\beta_0 \rightarrow 0} \frac{K}{\lambda_{\mathrm{g}}} \sqrt{\frac{1-\beta_0}{1+\beta_0}}=\frac{K}{\lambda_{\mathrm{g}}}=\frac{1.605 \mathrm{MV}}{\lambda_{\mathrm{g}}},          (13.316)

indicating that microwaves with an electric field amplitude \left({E}_z\right)_0 > 1.605 MV/λ_g would be able to catch stationary or low kinetic energy free electrons and accelerate them to speed of light c without any velocity modulation.
The β_0 → 0 limit, of course, begs the question on whether or not this is feasible in practice. Standard clinical linacs operate at a microwave frequency ν of 2856 MHz corresponding to microwave wavelength λ_g = 10.5 cm and a limit in (13.316) of 1.53\times 10^7 V/m. For miniature waveguides used in specialized equipment, such as CyberKnife and Tomotherapy, the operating frequency ν is 10^4 MHz corresponding to λ_g = 3 cm and a limit in (13.316) of 5.35\times 10^7 V/m. Currently, microwaves of electric field amplitudes exceeding 10^7 V/m are about an order of magnitude larger than the levels available from commercial microwave power sources (klystrons and magnetrons). However, during the past four decades of commercial development of clinical linacs, the electron gun potentials have been steadily dropping from about 150 kV to about 25 kV as a result of steady improvement in design of microwave power sources.

(2) The lower limit of [({E}_z)_0]_{min} is attained when β_0 → 1 corresponding to υ_0 = c

\lim _{\beta_0 \rightarrow 1}\left[\left({E}_z\right)_0\right]_{\min }=\lim _{\beta_0 \rightarrow 1} \frac{K}{\lambda_{\mathrm{g}}} \sqrt{\frac{1-\beta_0}{1+\beta_0}}=0 .          (13.317)

This limit is obviously trivial, since once the electron travels with velocity c there is no need for a further acceleration, at least not in the range of clinically relevant electron kinetic energies.

(e) A sketch of [({E}_z)_0]_{min} against β_0 for microwave frequency of ν = 2856 MHz (λ_g = 10.5 cm) is prepared by using the capture condition of (13.314) and calculating several points of [({E}_z)_0]_{min} = f (β_0) for 0 ≤ β_0 ≤ 1 in β_0 increments of 0.2. Results of the calculation are shown in Table 13.7 and Fig. 13.14 noting that the upper (β_0 → 0) and lower (β_0 → 1) limits in \left({E}_{z0}\right)_{min} that were discussed in (d) are also included in the table.