Medical Applications of Radiation Fields

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Esame Medical Applications of Radiation Fields

Facoltà Ingegneria dei processi industriali

Dal corso del Prof. Medicina Prof.

Università Politecnico di Milano

Publisher LGaravelli96

A.A. 2021-2022

82 pagine

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Equazioni di Maxwell

E E B̃ ̃ ̃B1 1̃(r ) z z zθ and= =2r r r̃ ̃t ̃ ̃tcwhere c is the light velocity in vacuum. The first equation can be written asẼB1 1̃ zB θ( +r )=θ 2r r t̃ ̃cand, by differentiating with respect to t: 22 Ẽ1 B B 1̃ ̃ zθ θ( +r )= 2 2r t t̃ ̃r ̃ c t̃while the differentiation with respect to r the second equation we get2 2Ẽ B̃z θ=2 t̃r ̃r̃By inserting this result in the former equation, we obtain2d E dE(r ) 21 ωz z E+ + (r )=0z2 2r drdr cwhich is a special form of Bessel’s equation, whose solution can be expressed in terms of the zero-orderBessel’s functions J (k r) and Y (k r), which are shown below.0 n 0 nFunction Y can be eliminated for the condition requiring that E must be finite on the cylinder axis. The0 zsolution is j ωtE r ,t J r e( )=E (k )zn 0n 0 nwhere E is the value of the electric field on the z-axis. The second boundary condition requires that the0nelectric field

parallel to the metallic wall at r=R must be zero (E (R ,t)=0) if the wall material is a good conductor. This implies that only some peculiar values of k give valid solutions. The allowable k values are determined by the zeros of the function J, which are listed in the table below.

Therefore the mode at the lowest frequency will be associated to the first zero of function J; in vacuum:

f = 2.405 * π * ω / c

Therefore, for R = 30 cm, the frequency f is of the order of 400 MHz, an appropriate frequency for RF power sources. The angular frequency ω is linked to the wave number k through the relation, in vacuum:

k * ω = 2 * π * f / c

from which c^2 * f * k * π = 2 * π * f / c

and therefore: 2 * π * k = n * λ

The magnetic field for the various modes can be calculated with Maxwell's equation B = -∇×E. As it was assumed, the magnetic field is directed along the θ direction. By assuming the same time profile of the electric field

and it is equal to zero for transverse modes. The TE modes have a longitudinal electric field component and a transverse magnetic field component, while the TM modes have a transverse electric field component and a longitudinal magnetic field component. The azimuthal mode (m) determines the number of field variations around the circumference of the cavity, while the radial mode (n) determines the number of field variations along the radial direction. The longitudinal mode (p) determines the number of field variations along the longitudinal direction.which is equal to zero in the example described above since E is constant along the z-direction.
zTM are particularly appropriate for particle acceleration. The longitudinal electric field is uniform along the 0n0 beam direction and its value is maximum on the beam axis. The transverse magnetic field is zero on the beam axis and this is important for electron acceleration where transverse magnetic fields can deflect the beam. TM modes with longitudinal wave numbers different from zero (p≠0) are characterized by axial electric fields of a shape Ez(p,x) ∝ sin(π/d)z and the acceleration of article traversing the cavity is lower. For n>1, the cavity can be thought as subdivided into n LC interactive resonant circuits. The capacitance and the inductance of each equivalent circuit are lowered by about a factor 1/n, and therefore the resonance frequency of this combination of elements is increased by about a factor n. Usually, resonant cavities are in copper or in stainless steel.

with electro-deposited copper for the highest conductivity. Nonetheless, the effects of resistivity are significant because of the large reactive current. Resistive energy losses for current flowing in the walls are concentrated in the inductive regions of the cavity. Therefore, the RLC circuit is a good approximation of an imperfect cavity.

A cylindrical cavity can provide a variety of resonant modes, generally at a higher frequency than that of the fundamental acceleration mode. Usually, the higher order modes are undesired, since they do not contribute to particle acceleration leading to energy waste.

Waveguides

Resonant cavities possess a finite extension along the axial direction. The electromagnetic waves are reflected at the axial boundaries, generating standing waves patterns that constitute the resonance modes. In this section, the cavity boundaries will be neglected for studying the electromagnetic oscillations traveling along the axial direction. A structure containing a propagating

is called a waveguide. They transport electromagnetic energy and are often used in particle accelerators to couple the power generated by a microwave source to a resonant cavity. Additionally, particle beams can be transported in a waveguide by synchronizing it with the wave phase velocity to accelerate it. Waveguides used for particle acceleration should provide slow waves with a phase velocity equal to or slower than the speed of light. Waveguide-based linear accelerators are used to accelerate electron beams for radiation therapy. Let's consider metallic structures with a uniform cross section and infinite length along the z direction. In particular, let's consider a cylindrical waveguide, which is a simple hollow pipe. Single frequency waves are transported in the guide through fields behaving as exp[j(ωt-kz)] or exp[j(ωt+kz)], so that the electromagnetic oscillations move along the waveguide with a phase velocity equal to ω/k. The waveguides show the following characteristics: - the phase velocity varies with the frequency. A structure with frequency-dependent phase velocity exhibits dispersion. - low frequency waves cannot propagate in a waveguide. The limiting frequency is called cut-off frequency. - the phase velocity in a hollow waveguide is higher than the light velocity. This does not violate the relativity principle, since information can only be extracted through frequency or amplitude modulation. - The propagation velocity of frequency modulations is the group velocity, which is always lower than the light velocity in a waveguide. Let us consider a wave propagating in the TM mode, so that only the electric fields possess a longitudinal component. As we saw earlier, we will concentrate on the TM because it possesses the optimum field variations for particle acceleration; the mathematical methods can be easily extended to other modes. The oscillation modes in azimuthal symmetry in a cylindrical guide can

be determined by solving the field equations. Let the electric and the magnetic field behave as

E(r, θ, z, t) = E(r, θ) exp[jt - kz(r, θ)]

B(r, θ, z, t) = B(r, θ) exp[jt - kz]

θ(r, θ) exp[(ω

Considering these forms and assuming that there are no free charges and currents into the wave guide, Maxwell's equations can be written as

jB∇ × E = -ωjE∇ × B = -ωεμ

These equations can be combined in order to obtain the following wave equations:

∇²E = -k²E∇²B = -k²B

where ωk = ωεμ = √0 v is called free-space wavenumber. This quantity can be linked to the wavelength λ of the electromagnetic waves in the cavity filling material without borders (i.e. an infinite cylinder) by

2πk = 0λ

In principle both wave equations can be solved for the three components of E and B. This procedure, however, is complicated by the boundary conditions which should be

Fulfilled at the wall radius: E_B(R)=0, B_⊥=0

Issues arise since these conditions couple equations for different components, hence an organized approach is needed in order to make the calculation tractable.

Let the solutions with azimuthal symmetry be considered only. Thus, setting ∂/∂θ=0, the components of Maxwell's equations results to be:

2kkE_jB₀=-ω_jkB_jE=-(ω/θ)r rθ1 E_θ(r) 2kB₁θ(r)j B_θ=-ω₀θj(E=-)z and r rθ(ω/θ)zr rθE_θ2B_kzθjkE_–jB₋=-(ω/θ)z 0jkB_–j(Eθr−=−)r(ω/θ)θr rθ

These equations can be handled algebraically so that the transverse fields are proportional to the derivatives of the longitudinal components, thus resulting in:

E_Bθ=−(ω/θ)r rθE_jk, B_jk=−r r²2k²−k²0 0B_Eθ=−(ω/θ)z zr rθE_jk, B_jk=−(ω/θ)

_θ2 2 2 2k k−k −k0 0It should be noted that no solution exists if both B and E are equal to zero and therefore a waveguide cannotz zprovide a TEM mode. These equations suggest a method for the simplification of the boundary conditions.Solutions are subdivided into two categories: waves with E =0 and waves with B =0. The former is a TE wavez zand the latter is a TM wave. The former type possesses transversal components B and E ., and the onlyr θmagnetic component perpendicular to the metallic wall is B . The following simple boundary condition isrobtained by setting B =0 on the walls:r ( )B_z =0r_R 0from which follows that E and B R(R )=0 ( )=0θ 0 r 0The wave equation for the B axial component can be solved with the above boundary condition. After havingdetermined B , the other components can be calculated through the expressions above.zFor TM modes, the transversal components are E and B

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