In a clinical linear accelerator electrons are generated by an electron gun and injected into the acceleration waveguide with a relatively low kinetic energy (EK)0 between 20 keV and 150 keV (depending on the gun design). At the end of the acceleration process the electrons exit the acceleration

Question

In a clinical linear accelerator electrons are generated by an electron gun and injected into the acceleration waveguide with a relatively low kinetic energy [latex]\left(E_K ight)_0[/latex] between 20 keV and 150 keV (depending on the gun design). At the end of the acceleration process the electrons exit the acceleration waveguide with kinetic energy [latex]E_K[/latex] between 4 MeV and 25 MeV (depending on waveguide design). Most modern waveguides do not use phase velocity modulation for capturing electrons injected from the gun into the acceleration waveguide; rather, they rely on the capture condition and employ sufficiently high electric field amplitude [latex]{E}_{z0}[/latex] of the accelerating radio-frequency (RF) fields to capture the injected electrons and accelerate them to the desired final kinetic energy.

(a) Calculate the minimum injection kinetic energy [latex]\left[\left(E_{\mathrm{K}} ight)_0 ight]_{\min }[/latex] of electrons in an acceleration waveguide with constant phase velocity [latex]υ_{ph}[/latex], electric field amplitude [latex]\left(E_z ight)_0 = 8 imes 10^6[/latex] V/m, and microwave frequency ν = 2856 MHz.
(b) Express the capture condition in the form [latex]\left[\left({E}_z ight)_0 ight]_{\min }[/latex] as a function of the injected electron kinetic energy [latex]\left(E_K ight)_0[/latex] and:

(1) Prepare a table listing [latex]\left[\left({E}_z ight)_0 ight]_{min}[/latex] of the capture condition (see Prob. 287) for microwaves of frequency ν = 2856 MHz and the following kinetic energies [latex]\left(E_K ight)_0[/latex] in MeV of an electron injected into the acceleration waveguide: 001, 0.01, 0.1, 1, 10, and 100.
(2) Determine two limits of interest on [latex]\left[\left({E}_z ight)_0 ight]_{min}[/latex] for [latex]\left(E_K ight)_0 → 0\ and\ \left(E_K ight)_0 → ∞[/latex].
(3) Plot the tabulated data as solid circle data points on a graph of [latex]\left[\left({E}_z ight)_0 ight]_{min}\ against\ \left(E_K ight)_0\ with\ \left[\left({E}_z ight)_0 ight]_{min}[/latex] on the ordinate axis (linear scale) and [latex]\left(E_K ight)_0[/latex] on the abscissa axis (logarithmic scale). Also plot on the graph with an open circle the result of the [latex]\left[\left(E_K ight)_0 ight]_{min}[/latex] calculation carried out in (a).

(c) Calculate the minimum electric field amplitude [latex]\left[\left({E}_z ight)_0 ight]_{min}[/latex] required for an electron acceleration waveguide operated with constant phase velocity [latex]υ_{ph}[/latex], final electron kinetic energy of 6 MeV, electron injection kinetic energy [latex]\left(E_K ight)_0[/latex] of 20 keV, and microwave frequency [latex]ν = 10^4[/latex] MHz.
(d) Comment on the dependence of the minimum electric field amplitude [latex]\left[\left({E}_z ight)_0 ight]_{min}[/latex] on: (1) wavelength [latex]λ_g[/latex] and frequency ν of microwaves used for acceleration of electrons with no velocity modulation and (2) velocity [latex]β_0[/latex] and kinetic energy [latex]\left(E_K ight)_0[/latex] of electrons injected into the acceleration waveguide for acceleration without velocity modulation of microwaves.

Accepted Answer

(a) The minimum injection kinetic energy [latex]\left(E_K\right)_{min}[/latex] is calculated from the capture condition (13.314) expressed in Prob. 287 as follows

[latex]\left[\left({E}_z\right)_0\right]_{\min }=\frac{K}{\lambda_{\mathrm{g}}} \sqrt{\frac{1-\beta_0}{1+\beta_0}},[/latex] (13.318)

where

[latex]\left[\left({E}_z\right)_0\right]_{min}[/latex] is the minimum microwave electric field amplitude required for capturing an electron injected into the acceleration waveguide with initial normalized velocity [latex]β_0[/latex].
K is the capture constant of the electron [latex]K = πm_ec^2/e[/latex] = 1.605 MV.
[latex]λ_g[/latex] is the wavelength of the microwaves used for electron acceleration in the waveguide.
[latex]β_0[/latex] is the normalized initial (injection) velocity of the electron injected into the waveguide.

In this problem, [latex]\left({E}_z\right)_0[/latex] in (13.318) is known [latex]\left(8\times 10^6\ V/m\right)[/latex] and we are looking first for the minimum required normalized velocity [latex]\left(β_0\right)_{min}[/latex] and then for minimum required kinetic energy [latex]\left(E_K\right)_{min}[/latex] that satisfies (13.318) for the given electric field amplitude [latex]\left({E}_z\right)_0[/latex]. We write (13.318) as

[latex]\left({E}_z\right)_0=\frac{K}{\lambda_{\mathrm{g}}} \sqrt{\frac{1-\left(\beta_0\right)_{\min }}{1+\left(\beta_0\right)_{\min }}} \text { or }\left[\frac{\lambda_{\mathrm{g}}}{K}\left({E}_z\right)_0\right]^2 \equiv \kappa^2=\frac{1-\left(\beta_0\right)_{\min }}{1+\left(\beta_0\right)_{\min }} \text {, }[/latex] (13.319)

where we introduce a new constant κ for the problem at hand

[latex]\kappa=\frac{\lambda_{\mathrm{g}}\left({E}_z\right)_0}{K}=\frac{c\left(E_z\right)_0}{v K}=\frac{\left(3 \times 10^8 \mathrm{~m} / \mathrm{s}\right) \times\left(8 \times 10^6 \mathrm{~V} / \mathrm{m}\right)}{\left(2856 \times 10^6 \mathrm{~s}^{-1}\right) \times\left(1.605 \times 10^6 \mathrm{~V}\right)}=0.524 .[/latex] (13.320)

Solving (13.319) for [latex]\left(β_0\right)_{min}[/latex] we get

[latex]\left(\beta_0\right)_{\min }=\frac{1-\kappa^2}{1+\kappa^2}=\frac{1-0.524^2}{1+0.524^2}=0.569[/latex] (13.321)

for the minimum normalized velocity [latex]\left(β_0\right)_{min}[/latex] of an electron injected into the acceleration waveguide in which the accelerating microwave electric field amplitude [latex]\left({E}_z\right)_0[/latex] is 8 MV/m.
A [latex]\left(β_0\right)_{min}[/latex] of 0.569 corresponds to the following minimum kinetic energy [latex]\left(E_K\right)_{min}[/latex] of the electron injected into the acceleration waveguide

[latex]\begin{aligned} \left(E_{\mathrm{K}}\right)_{\min } & =m_{\mathrm{e}} c^2\left(\frac{1}{\sqrt{1-\left(\beta_0\right)_{\min }^2}}\right)=(0.511 \mathrm{MeV}) \times\left(\frac{1}{\sqrt{1-0.569^2}}-1\right) \ & =0.110 \mathrm{MeV}\quad (13.322) \end{aligned}[/latex]

Thus, the minimum required kinetic energy [latex]\left(E_K\right)_{min}[/latex] that an electron must possess to be captured by microwaves of frequency ν = 2856 MHz and electric field amplitude [latex]\left(E_z\right)_0[/latex] = 8 MV/m upon its injection from the electron gun into the acceleration waveguide is 110 keV.

(b) The minimum electric field amplitudes [latex]\left[\left({E}_z\right)_0\right]_{min}[/latex] of ν = 2856 MHz microwaves that are just high enough to capture electrons of a given kinetic energy [latex]\left(E_K\right)_0[/latex] injected into an acceleration waveguide are determined using the capture condition (T13.110) expressed in the form [latex]\left[\left({E}_z\right)_0\right]_{min}[/latex] as a function of [latex]\left(E_K\right)_0[/latex] rather than [latex]β_0[/latex].
The modification of the capture condition from the [latex]β_0\ to\ \left(E_K\right)_0[/latex] dependence is simple, since [latex]β_0\ and\ \left(E_K\right)_0[/latex] are related through the following relativistic expressions (T2.7)

[latex]\left(E_{\mathrm{K}}\right)_0=m_{\mathrm{e}} c^2\left[\frac{1}{\sqrt{1-\left(\beta_0\right)^2}}-1\right] \quad \text { or } \quad \beta_0=\sqrt{1-\frac{1}{\left[1+\frac{\left(E_{\mathrm{K}}\right)_0}{m_{\mathrm{e}} c^2}\right]^2}},[/latex] (13.323)

where [latex]m_ec^2[/latex] is the rest energy of the electron (0.511 MeV).
Insertion of (13.323) into the capture condition (13.318) results in the following expression for the capture condition

[latex]\left[\left({E}_z\right)_0\right]_{\min }=\frac{K}{\lambda_{\mathrm{g}}} \sqrt{\frac{1-\sqrt{1-\frac{1}{\left[1+\frac{\left(E_{\mathrm{K}}\right)_0}{m_{\mathrm{e}} c^2}\right]^2}}}{1+\sqrt{1-\frac{1}{\left[1+\frac{\left(E_{\mathrm{K}}\right)_0}{m_{\mathrm{e}} c^2}\right]^2}}}}[/latex] (13.324)

(1) Equation (13.324) was used to calculate [latex]\left[\left({E}_z\right)_0\right]_{min}[/latex] as a function of [latex]\left(E_K\right)_0[/latex] data presented in Table 13.8 and shown with solid circles in Fig. 13.15. The limit of [latex]\left[\left({E}_z\right)_0\right]_{min}\ for\ \left(E_K\right)_0 → 0\ is\ K/λ_g[/latex] and the limit of [latex]\left[\left({E}_z\right)_0\right]_{min}\ for\ \left(E_K\right)_0[/latex] → ∞ is 0.

(2) The two limits of (13.324) for [latex]\left(E_{\mathrm{K}}\right)_0 \rightarrow 0 \text { and }\left(E_{\mathrm{K}}\right)_0 \rightarrow \infty[/latex] corresponding to limits [latex]β_0 → 0\ and\ β_0 → 1[/latex], respectively, are

[latex]\begin{aligned} \lim _{\left(E_{\mathrm{K}}\right)_0 \rightarrow 0}\left[\left({E}_z\right)_0\right]_{\min } & =\lim _{\beta_0 \rightarrow 0}\left[\left({E}_z\right)_0\right]_{\min }=\lim _{v_0 \rightarrow 0}\left[\left({E}_z\right)_0\right]_{\min } \ & =\frac{K}{\lambda_{\mathrm{g}}}=\frac{1.605 \times 10^6 \mathrm{~V}}{10.5 \times 10^{-2} \mathrm{~m}}=15.3 \mathrm{MV} / \mathrm{m}\quad (13.325) \end{aligned}[/latex]

and

[latex]\lim _{\left(E_{\mathrm{K}}\right)_0 \rightarrow \infty}\left[\left({E}_z\right)_0\right]_{\min }=\lim _{\beta_0 \rightarrow \infty}\left[\left({E}_z\right)_0\right]_{\min }=\lim _{v_0 \rightarrow \infty}\left[\left({E}_z\right)_0\right]_{\min }=0 .[/latex] (13.326)

(3) The data of Table 13.8 for 2856 MHz microwaves are plotted in Fig. 13.15 with solid circles on a semi-logarithmic plot of the form [latex]\left({E}_{z0}\right)_{min}[/latex] on the ordinate (linear scale: y axis) against [latex](E_K)_0[/latex] on the abscissa (logarithmic scale: x axis). The open circle on the graph represents the result of the [latex]\left(E_K\right)_0[/latex] calculation for [latex]\left[\left({E}_z\right)_0\right]_{min} = 8\times 10[/latex] MV/m.

(c) The minimum electric field amplitude [latex]\left[\left({E}_z\right)_0\right]_{\min } \text { of } v=10^4 \mathrm{MHz}\left(\lambda_{\mathrm{g}}=3 \mathrm{~cm}\right)[/latex] microwaves that is just high enough to capture electrons of kinetic energy [latex]\left(E_K\right)_0[/latex] = 25 keV = 0.025 MeV injected into a miniature acceleration waveguide is determined using the capture condition (13.324) as follows

[latex]\left[\left({E}_z\right)_0\right]_{\min }=\frac{K}{\lambda_{\mathrm{g}}} \sqrt{\frac{1-\sqrt{1-\frac{1}{\left[1+\frac{\left(E_{\mathrm{K}}\right)_0}{m_{\mathrm{e}} c^2}\right]^2}}}{1+\sqrt{1-\frac{1}{\left[1+\frac{\left(E_{\mathrm{K}}\right)_0}{m_{\mathrm{e}} c^2}\right]^2}}}}\[/latex]

[latex]\begin{aligned} & =\frac{1.605 \times 10^6 \mathrm{~V}}{3 \times 10^{-2} \mathrm{~m}} \times \sqrt{\frac{1-\sqrt{1-\frac{1}{\left[1+\frac{20 \times 10^3}{0.511 \times 10^6}\right]^2}}}{1+\sqrt{1-\frac{1}{\left[1+\frac{20 \times 10^3}{0.511 \times 10^6}\right]^2}}}} \ & \end{aligned}\ \begin{aligned} & =\left[53.5 \times 10^6 \frac{\mathrm{V}}{\mathrm{m}}\right] \times \sqrt{\frac{1-0.272}{1+0.272}} \ & =\left[53.5 \times 10^6 \frac{\mathrm{V}}{\mathrm{m}}\right] \times 0.757=40 \mathrm{MV} / \mathrm{m} .\quad (13.327) \end{aligned}[/latex]

(d) As seen from the capture condition (13.318) and the modified capture condition (13.324), the minimum electric field amplitude [latex]\left[\left({E}_z\right)_0\right]_{min}[/latex] is inversely proportional to microwave wavelength [latex]λ_g[/latex] or proportional to microwave frequency ν since [latex]ν = c/λ_g[/latex]. Thus, for the same electron injection velocity [latex]β_0[/latex] or same injection kinetic energy [latex]\left(E_K\right)_0[/latex] the minimum electric field amplitude [latex]\left[\left({E}_z\right)_0\right]_{min}[/latex] increases with microwave frequency ν. This means that a miniature waveguide operated in the X band at [latex]10^4[/latex] MHz [latex]\left(λ_g = 3\ cm\right)[/latex] has by factor of 10.5/3 = 3.5 larger [latex]\left[\left({E}_z\right)_0\right]_{min}[/latex] than a standard linac acceleration waveguide operating in the S band at a frequency of ν = 2856 MHz corresponding to a wavelength of [latex]λ_g[/latex] = 10.5 cm.
As also seen from the capture condition of (13.318) and modified capture condition of (13.324), for a given microwave wavelength [latex]λ_g[/latex] or frequency ν, the maximum in [latex]\left[\left({E}_z\right)_0\right]_{\min }=K / \lambda_{\mathrm{g}} \text { occurs as } \beta_0 \rightarrow 0 \text { corresponding to }\left(E_{\mathrm{K}}\right)_0 \rightarrow 0[/latex]. Furthermore, as [latex] β_0[/latex] [i.e., [latex] (E_K)_0[/latex] ] increase from zero, [latex]\left[\left({E}_z\right)_0\right]_{min}[/latex] decreases from [latex]K/λ_g[/latex] and approaches zero as [latex]β_0[/latex] → 1 corresponding to [latex]\left(E_K\right)_0[/latex] → ∞.

Table 13.8 Minimum electric field amplitude $\left[\left({E}_z\right)_0\right]_{min}$ of ν = 2856 MHz microwaves against kinetic energy $\left(E_K\right)_0$ calculated using (13.324) for various incident kinetic energies $\left(E_K\right)_0$ of electrons injected into an acceleration waveguide
$\left(E_K\right)_0$ (MeV)	0	0.001	0.01	0.1	1.0	10	100	∞
$\left[\left({E}_z\right)_0\right]_{min}$ (MV/m)	15.3	14.4	12.6	8.26	2.66	0.38	0.004	0

Question 13.11.Q2: In a clinical linear accelerator electrons are generated by ......

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