With the development of high energy physics, medicine, materials science and molecular biology, the urgent needs of effectively promoting new accelerate principle and the study of the method of new acceleration ask a new generation of compact accelerator with high gradient and low losses.

At present, we use microwave to accelerate particles usually, but as we know, the high energy stored in the microwave cavity is limited by the metal surface breakdown and power losses at conducting cavities and waveguides. In recent years, in the areas of advanced accelerator research and development, there have been many methods which have been proved to accelerate the electron to very high energy. For example, laser wake-field accelerator (LWFA [1]), plasma wake-field accelerator (PWFA [2]), plasma beat-wave accelerator (PBWA) [3], dielectric wake-field accelerator (DWA) [4], inverse free electronic laser (IFEL) [5], inverse Cherenkov accelerator (ICA) [6], the vacuum laser accelerators(VLA) [7], and so on. But all the above need a very high energy laser beam(magnitudes of TW) and are limited to very short distances because of keeping synchronous.

In this paper, a composite resonator is presented for linacs, whose accelerating units are comprised of periodically shaped cavities with a smaller dielectric sphere located at a spherical conducting cavity center [8]. See Figure 1. We know that when a plane wave is scattered on a limited size dielectric object, the structural resonances can be excited in the object, e.g., whisper gallery modes (WGM). Because of the resonance, the amplitudes of the electromagnetic fields in the dielectric sphere and its vicinity are very high, the maximum value exceeds by several orders of that in typical resonant linear accelerators. A particular example of the WGM resonance shows that the accelerating fields are almost 100 times stronger than those in the incident wave [9]. So take usage of this structure to accelerate charged particles is very effective.

For the conventional accelerators, the strength of fields for the conventional accelerators is limited by electric breakdown and power losses at conducting cavities and waveguides. Compared to the existing particle accelerators, the skin current losses are reduced without engaging superconductivity and cryogenic systems for this new construction. Especially because all field components at the metallic wall are either zero or very small in this proposed spherical cavity through our simulation, one can expect the cavity to be less prone to electrical breakdowns than the traditional cavity. In the following sections a new type of accelerating structure was analyzed and calculated.

For the composite resonator structure, the accelerating fields had been discussed in standard textbooks on high energy particle accelerators, e.g. Jackson Classic Electrodynamics [10]. Because the structure is symmetric, the TE field inside a spherical cavity with a dielectric sphere in the spherical coordinate system has the following form

{ E l m ( r , t ) = E l m ( r , ω ) e − i ω t E l m ( r , ω ) = E t ( k r ) X l m ( θ , ϕ ) ,                                                                   ( 1 ) B t l m ( r , t ) = B t l m ( r , ω ) e − i ω t B t l m ( r , ω ) = B t ( k r ) n × X l m ( θ , ϕ ) ,                                                           ( 2 ) B r l m ( r , t ) = B r l m ( r , ω ) e − i ω t B r l m ( r , ω ) = B r ( k r ) n Y l m ( θ , ϕ ) ,                                                                 ( 3 )

where k = ω/c, and ω is the angular frequency, ( r , θ , ϕ ) are spherical coordinates, the complex dielectric permittivity is ϵ , Y l m ( θ , ϕ ) is spherical harmonics and X l m ( θ , ϕ ) is vector spherical harmonics as defined in Jackson’s Classical Electrodynamics [10], n = r / r , and l is a positive integer related to the integer m by – l ≤ m ≤ l .

Considering boundary conditions that the tangent component of the electric field must be zero at the ideal metallic wall and must be regular at the cavity center, from Jackson’s Classical Electrodynamics we can assume that the form of the tangent component is as follows [8]:

E t ( k r ) = N l × { A l j l ( ϵ k r )                       0 ≤ r < a j l ( k r ) j l ( k b ) − y l ( k r ) y l ( k b )           a ≤ r ≤ b (4)

wherein j l ( ρ ) ≡ π 2 ρ J l + 1 2 ( ρ ) and y l ( ρ ) ≡ π 2 ρ Y l + 1 2 ( ρ ) are spherical Bessel and Neumann functions, and a and b are radius of the dielectric sphere and the outer cavity.

From Maxwell’s equation, we know

∇ × E = i k B (5)

From Equations (1) (4) and (5) we get

B t ( k r ) = − i N l k r × { A l j l D ( ϵ k r )                               0 ≤ r < a j l D ( k r ) j l ( k b ) − y l D ( k r ) y l ( k b )             a ≤ r ≤ b (6)

B r ( k r ) = l ( l + 1 ) k r E t ( k r ) (7)

where j l D ( ρ ) = d d ρ ( ρ j l ( ρ ) ) and y l D ( ρ ) = d d ρ ( ρ y l ( ρ ) ) are derivatives of the Riccati-Bessel and Riccati-Neumann functions.

Using the fact that the tangent electric field and magnetic induction must be continuity at the boundary of the dielectric sphere, we can infer the following equation

j l ( ϵ k a ) [ j l D ( k a ) j l ( k b ) − y l D ( k a ) y l ( k b ) ] − j l D ( ϵ k a ) [ j l ( k a ) j l ( k b ) − y l ( k a ) y l ( k b ) ] = 0 (8)

From this equation, we can inform the eigen-frequencies of the cavity modes ω = ω l from k = k l . And with the choice of k l , A l can be obtained. On the basis of the above equations, assuming the electrons transit length through the cavity is the same as that of SLAC pill box cavity d = 4 cm, we take four groups of a andb which satisfy the electrons transit length in the composite resonator d = 4 cm to calculate the resonant frequenciesf(l) with MATLAB, the result is shown in Figure 2.

Figure 3 shows that all field components at the metallic wall are either zero or very small, so one can expect the cavity to be less prone to electrical breakdowns than the traditional cavity.

Let a = 0.708 cm, b = 2.124 cm, we simulated the electric field gradient distribution curve along the direction of electron motion, see Figure 4. we observe that the maximum value of the electric field is distribute very closely vicinity of the dielectric sphere.

Figure 5 is the tangential electric field along the radial distribution, which is simulated with CST, it shows that the result match very well with that of the calculation with Matlab.

Figure 6 shows the simulated result of the electric field distribution in the composite resonator, we can see the electric field along the trajectory is very large.

As for the Quality Factor, it is an important parameter for any linear accelerators, which is defined as

Q = ω 0 U P = 2 π U T 0 P (9)

In which ω 0 is the resonance angular frequency of the ideal cavity, in which cavity σ = ∞ , ϵ ″ = 0 , T₀ is the corresponding resonance period, P is time averaged cavity power loss, and U is the time energy stored in the cavity

U = 1 16 π ∫ V ( ϵ ′ | E | 2 + | H | 2 ) d V (10)

In the above expression, μ = 1 is used.

The power loss caused by the skin current in the metallic wall bounded by the surface S is given by the following expression

P m e t = c 2 32 π 2 σ δ ∫ s | H | 2 d s (11)

where

δ = c 2 π σ ω 0 (12)

σ is conductivity of the wall, δ is the skin depth, and the normal component of the magnetic field intensity H is vanishing on the surface S (ideal cavity), so

H = H t

The Q factor of the cavity related to the loss in the metallic wall is given by

Q m e t = ω 0 U P m e t (13)

Because at resonance, the averaged electric and magnetic energy stored in the cavity are equal, we can obtain

Q m e t = 2 δ ∫ V | B | 2 d v ∫ s | B | 2 d s (14)

For μ = 1 , H = B .

For traditional cylindrical cavity of radius R_c and height h,

Q c = 2.405 2 π σ ω 0 1 1 + R c / h (15)

For the SLAC pill box cavity, the typical values are R c = h = d = 4   cm , σ = 5.294 × 10 17     s − 1 ,then Q c = 1.633 × 10 4 can be obtained. The corresponding values for spherical cavities with ideal dielectric spheres and the same values of d and σ reach much larger values, see Figure 7.

In the presence of the dielectric sphere, we need to deal with the losses due to an imperfect dielectric specified by ϵ ″ = Im ϵ ′ ≠ 0 , so Im k ≠ 0 , which means that ω = ω ′ + i ω ″ is complex.

For the fields given by (1-7), we can get

U ( t ) = U ( t = 0 ) e 2 ω ″ t

So

P d i e l = − d U d t = − 2 ω ″ U

And the corresponding quality will be

Q d i e l = − ω 0 − 2 Im ω (16)

This value is of the order of tan δ − 1 = ϵ ′ / ϵ ″ , and It is approximately l independent. With the ultra-small losses dielectric in [9], ϵ ′ = 10 , ϵ ″ = 10 − 6 , Q d i e l = 10 7 .

The total quality factor can be obtained by the relation

1 Q s = 1 Q m e t + 1 Q d i e l (17)

Values of Q_s versus l for three spherical cavities with dielectric spheres and are shown in Figure 7.

From the calculation results shown in Figure 7 we can see that, the quality factor in the resonant cavity with a dielectric sphere located at a spherical conducting cavity center are much larger than that in SLAC whose Q c = 1.633 × 10 4 . Furthermore, the quality factor of such a resonator only depends on the losses in the dielectric. For existing ultra-low loss dielectrics, Q can be three orders of magnitude better than obtained in existing cylindrical cavities.

The above-described details show that, with the use of the composite cavity, we can obtain high accelerating gradient and low loss, especially because all field components at the metallic wall are either zero or very small in this proposed spherical cavity, one can expect the cavity to be less prone to electrical breakdowns than the traditional cavity. It shows potential possibilities of significant increase of acceleration.

The authors declare no conflicts of interest regarding the publication of this paper.

Tian, X.F., Meng, H.Y., Li, C.H. and Li, H.Y. (2022) Theoretical Study on Spherical Composite Accelerating Cavity. Journal of Electromagnetic Analysis and Applications, 14, 39-46. https://doi.org/10.4236/jemaa.2022.144004