_{1}

^{*}

The quantization of circuits has received to be rather attractive in domains of solid state—molecular—and biophysics, since the quanta referred to as Q-bits play a significant role in the design of the quantum computer and entangled structures. Quantized circuits cannot be applied without modifications, since the energy differences are not equidistant and the polarization of the excited states has to be accounted for having particular importance for the creation of virtual states. Applications of the presented theory are scanning methods in radiotherapy without multi-leaf collimators, which may be realized in tomo-scanning radiotherapy and in the keV domain, which provides a new design of CT. The problem of lateral scatter in the target and energy storage by heat production is significantly reduced by a multilayer system with focusing the impinging electrons at the walls and by a magnetic field. The verification of the Heisenberg-Euler scatter of crossing beams of 9 MV is a central problem of photon physics and can be solved by the new bremsstrahlung technique. A comparison with GEANT 4 Monte-Carlo data indicates that the presented method also works in the GeV domain, and a multi-target can improve the bremsstrahlung yield. GEANT 4 provides the spatial distribution, whereas the virtual oscillator states only show the created energy spectrum. In every case, the exploitation yield can be drastically improved by the superiority of the focused multitarget system compared to a single standard target, and the door to new technologies is opened.

The forced circuit oscillator in quantum theory, which contains the addition term of the form U(t) = U_{0}∙e^{i}^{∙ω∙t}, is significant in many disciplines of pure and applied physics, such as quantum electrodynamics, molecular and biophysics inclusive biorhythms [_{0}, and only a resistance term can overcome this problem. The translation of the resistance problem to quantum mechanics is not a trivial one, since the uncertainty relation must not be violated, and a nonlinear term in the Schrödinger equation must be accounted for [

Abbreviations: Inductivity of a coil: L; mutual magnetic coupling between coils: M; Capacitor: C; mutual electric coupling C_{I} by common dielectrics ε. If ε = 1: vacuum or air with C_{0} = C; in the presence of the electric coupling the capacitor reads C = C_{0}(1 + C_{I}). However, by virtue of the time-dependence of U_{0}(t) the dielectric factor ε assumes ε > 1 and may become a function of the frequency ω_{0}, i.e.: ε = ε(ω_{0}). The creation of bremsstrahlung can be founded by coupled harmonic oscillator processes, where besides the mutual magnetic coupling the electric coupling plays a significant role, since the role of virtual orbitals (oscillator states) is closely connected to the polarizability of the dielectric properties. The method even works for bremsstrahlung up to 9 GeV.

The following denominations and explanations used in

_{1}·e^{iωt} and U_{2}·e^{iωt}. In addition to the usually mutual magnetic coupling M the electric coupling C_{I} is included, which may assume either positive or negative values. The special case without electric and magnetic coupling and U_{1} = U_{2} = U_{0} yields two independent circuits. The indices of the charge can be omitted, and in classical electrotechnics Formula (1) plays a significant role:

L ⋅ Q ¨ + 1 C 0 ⋅ Q = U 0 ⋅ e i ⋅ ω ⋅ t → Q ¨ + ω 0 2 ⋅ Q = U 0 L ⋅ e i ⋅ ω ⋅ t } (1)

The well-known eigenfrequency of this oscillator is given by:

ω 0 2 = 1 L ⋅ C 0 (1a)

With the help of the “ansatz” Q = Q 0 ⋅ e i ⋅ ω ⋅ t the solution of Equation (1) assumes the shape:

Q 0 = 1 ω 0 2 − ω 2 ⋅ U 0 L (2)

As well-known, Equation (1) yields Equation (1a), if ω = ω_{0} and may only be overcome by an additional damping term, which in electrotechnics is an Ohm resistance R. An immediate translation of Equation (1) by taking account for damping to QM might require a nonlinear Schrödinger equation in order to incorporate damping as already mentioned. However, this problem can be circumvented: The principal aspect in the present study is the creation of bremsstrahlung according to Sections 2 and 3. The creation of bremsstrahlung is always connected with scatter of impinging electrons and heat production in the related medium (usually tungsten), yet the frequency-depending dielectric factor ε(ω) appearing in all formulas of this study will be assumed to be complex, where the imaginary part describes the losses by heat production and scatter. By that, we shall make use of the dispersion methods in classical and quantum optics, where similar tasks are treated, e.g., the absorption and emission of light.

In the succeeding section, it is a very essential feature that the dielectric constant ε in the capacitor C is itself depending on the eigenfrequency ω_{0}, i.e., ω_{0} = ω_{0}(ε), in order to remove the equidistant energy levels E_{n} = (n + 1/2)·ħ·ω_{0}, which are not adequate neither in low-energy molecular physics nor in radiation and high-energy physics.

This chapter is based on

L ⋅ Q ¨ 1 + M ⋅ Q ¨ 2 + 1 C ⋅ Q 1 − C I C ⋅ Q 2 = U 1 ⋅ e i ⋅ ω ⋅ t L ⋅ Q ¨ 2 + M ⋅ Q ¨ 1 + 1 C ⋅ Q 2 − C I C ⋅ Q 1 = U 2 ⋅ e i ⋅ ω ⋅ t } (3)

Using the substitutions q_{1} = Q_{1} + Q_{2}, q_{2} = Q_{1} − Q_{2}, U_{11} = U_{1} + U_{2}, U_{22} = U_{1} − U_{2}, λ_{1} = L + M, λ_{2} = L − M, C_{1} = C/(1 − C_{I}), C_{2} = C/(1 + C_{I}) and passing to the Lagrangean, which reads:

L = λ 1 2 ⋅ q ˙ 1 2 − 1 2 ⋅ C 1 ⋅ q 1 2 + U 11 ⋅ λ 1 ⋅ e i ⋅ ω ⋅ t + λ 2 2 ⋅ q ˙ 2 2 − 1 2 ⋅ C 2 ⋅ q 2 2 + U 22 ⋅ λ 2 ⋅ e i ⋅ ω ⋅ t (4)

In Equation (3), a negative capacitive coupling is assumed, yet a positive coupling only requires the substitutions C_{1} = C/(1 + C_{I}), C_{2} = C/(1 − C_{I}), and Equation (4) will not be changed.

The canonical momenta are given by:

P 1 = λ 1 ⋅ q ˙ 1 ; P 2 = λ 2 ⋅ q ˙ 2 (5)

By that, both Hamiltonians assume the shape:

H 1 = 1 2 ⋅ λ 1 ⋅ P 1 2 + λ 1 2 ⋅ ω 1 2 ⋅ q 1 2 − q 1 ⋅ U 11 ⋅ e i ⋅ ω ⋅ t H 2 = 1 2 ⋅ λ 2 ⋅ P 2 2 + λ 2 2 ⋅ ω 2 2 ⋅ q 2 2 − q 2 ⋅ U 22 ⋅ e i ⋅ ω ⋅ t } (6)

ω 1 2 = 1 λ 1 ⋅ C 1 ; ω 2 2 = 1 λ 2 ⋅ C 2 (6a)

The Schrödinger equation is only a slight modification of Equation (6). Therefore, we pass to the representation by creation—and annihilation operators:

P k ψ → ℏ i ∂ ∂ q k ψ ; ρ k = U k ⋅ ℏ 2 ⋅ λ k ⋅ ω k ; ξ k 2 = λ k ⋅ ω k ℏ ⋅ q k 2 k = 1 , 2 ; U k = U 11 ( k = 1 ) ) ; U k = U 22 ( k = 2 ) } (7)

With the help of Equation (7) the Schrödinger equation assumes the shape:

− 1 2 ∂ 2 ∂ ξ k 2 ψ k + 1 2 ξ k 2 ψ k − ρ k U k ψ k = i ⋅ ℏ ∂ ∂ t ψ k (7a)

In order to introduce creation and annihilation operators we use the following relations:

b k + = 1 2 ( − ∂ ∂ ξ k + ξ k ) ; b k = 1 2 ( ∂ ∂ ξ k + ξ k ) b k b l + − b l + b k = δ k l ( k , l = 1 , 2 ) } (7b)

By that, Equation (7a) becomes:

H k ⋅ ψ k = [ ℏ ω k ⋅ ( b k + b k + 1 2 ) − ρ k ⋅ ( b k + b k + ) e i ω t ] ψ k = i ℏ ∂ ∂ t ψ k (8)

The creation of bremsstrahlung in a target (usually tungsten is applied) is always connected with heat production and scatter in this medium. Therefore, we have to analyze sufficiently the co-lateral processes before we are able to apply Equations (6) - (8) to this task (Z: nuclear charge of the stopping medium, A_{N}: mass number, and ρ: density) and have a look at Bethe-Bloch equation:

− d E ( z ) d z = K m c 2 β 2 ⋅ [ ln ( 2 m c 2 β 2 E I ( 1 − β 2 ) ) + a s h e l l + a B a r k a s + a 0 v 2 + a B l o c h ] K = ( Z ρ / A N ) ⋅ 8 π q 2 e 0 4 / 2 m ; β = v c } (9)

v 2 = 2 E / m ; β I = 4 / E I ; E = ( 1 / β I ) exp ( − u / 2 ) (9a)

1 2 K ⋅ m ⋅ β 2 ∫ d z = ∫ d u e − u u + 2 α s + 2 α B a r k a s ( 4 ⋅ m / E I ) + p B e p B u / 2 + α 0 ( E I / 2 m ) e − u / 2 (9b)

With reference to the weighted ionization potential of the stopping medium we have checked the usual value E_{I} = 75.1 eV for water, which is tacitly assumed to be valid for other stopping media. For this purpose, we have used the ionization energy levels of the atomic shells for water, lead and tungsten determined by ICRU [_{ik} induced by collisions with external electrons. Now the average ionization potential is obtained by the weighted energy levels divided by the number of energy shells, which is Z = 10 for water, Z = 74 for tungsten, and Z = 82 for lead:

E I = 1 Z ⋅ ∑ i = 1 Z ∑ k = 1 ( k ≠ i ) Z E k ⋅ W i k (9c)

The condition k ≠ i must hold, since transitions can only occur by different energy levels; it should be added that Formula (9c) is a special case of the Pauli mater equation, and it provides some noteworthy results: For water, this formula yields E_{I} = 75.112 eV, which is in acceptable agreement with the assumed E_{I} = 75.1 eV. The results for lead and tungsten are more interesting, since these materials are used for radiation shielding and creation of bremsstrahlung. Thus, for tungsten we obtain E_{I} = 74.45 eV and for lead E_{I} = 84.92 eV. According to ICRU [^{2} and the relativistic mass dependence cannot be ignored.

The substitutions (9a) in Equation (9) yield the integration Formula (9b). The above Formula (9) is valid for protons and electrons, but the Barkas correction does not exist for projectile electrons, and, by setting α_{Barkas} = 0, the evaluation of the above Formula (9b) can be made easier. It is easy to verify that the correction terms prevent the singularity of the integration and by putting them to zero, the divergent integral would remain:

1 2 K ⋅ m ⋅ β 2 ∫ d z = ∫ d u e − u u (9d)

We use the operator calculus developed by Feynman [

[ A + B ] − 1 = A − 1 − A − 2 B + A − 3 B 2 − A − 4 B 3 + ⋯ + ( − 1 ) n A − n − 1 B n (9e)

The expression (9e) provides the integration of (9a) up to arbitrary order with finite values [

A practical problem in radiotherapy is to construct individual panels within an electron tube (lead allow, thickness: 1 cm), since electrons create bremsstrahlung at the panel surface.

R c s d a = C ⋅ q ⋅ E 0 ⋅ [ 1 − a ⋅ ( 1 − e − μ 0 ⋅ E 0 ) ] ⋅ 1 f E 0 = 1 C ⋅ a 1 ⋅ f ⋅ R c s d a ⋅ ( 1 + ε ⋅ e μ 1 ⋅ R c s d a + a 2 ⋅ e − β 2 ⋅ R c s d a 2 ) } (10)

q = 0.5285000251 ⋅ 1 f ; a = 0.7523823 ; μ = 0.0085004 ; Z w = 9.75 ; A w = 17.63 ; a 1 = 0.7570000003 ⋅ f ; a 2 = 0.0128751 ; β = 0.0035002 ; ε = 0.9939000094 ; μ 1 = 0.02639062433331674 } (10a)

According to Formula (10) the mean stand standard deviation of R_{csda} amounts to 0.008 cm and of E_{0} to 0.004 MeV, if compared with ICRU, which may be downloaded by ICRU [_{w} to A_{w}) the center-of-charge and center-of-mass of H_{2}O is more accurate. Please note that Equations (10, 10a) refer to water with C = 1. The correction factors C and C^{−1} have also to be accounted for, if one passes to high Z media with different values of E_{I}. Thus, for tungsten we have to use C = 0.992 and for lead C = 0.989. A further correction refers to the factor f, which is f = 1 for water (this makes sense, since water is used as the reference medium in the dosimetry), but by the fixations Z_{m}, A_{m} and ρ_{m} referring to the considered medium a general modification factor f has to be accounted for:

f = Z m ⋅ ρ m ⋅ A w A m ⋅ Z w (10b)

Based on Equations (10, 10a, 10b) the energy E(z) and the stopping power −dE/dzare given by:

E ( z ) = f C ⋅ a 1 ⋅ ( R c s d a − z ) ⋅ ( 1 + ε ⋅ e μ 1 ⋅ ( R c s d a − z ) + a 2 ⋅ e − β 2 ⋅ ( R c s d a − z ) 2 ) − d E d z = f C ⋅ [ a 1 ⋅ ( 1 + ε ⋅ e μ ( R c s d a − z ) + a 2 ⋅ e − β 2 ⋅ ( R c s d a − z ) 2 ) + a 1 ⋅ ( R c s d a − z ) ⋅ ( − μ ε ⋅ e μ ( R c s d a − z ) + a 2 ⋅ ( − 2 β 2 ( R c s d a − z ) e − β 2 ⋅ ( R c s d a − z ) 2 ) ) ] } (11)

A particular feature of Equation (11) is that the stopping power (−dE/dz) remains finite at the endpoint track of the electron (

− d E d z = f C ⋅ a 1 ⋅ ( 1 + ε + a 2 ) ( if z = R c s d a ) (11a)

With respect to the ranges of electron energies, we have restricted ourselves to 30 MeV, although we explicitly consider bremsstrahlung production of 6 MeV and 9 MeV electrons with tungsten targets, but this restriction is implied by two specific cases, whereas the further electron energies mainly serve to provide information on shielding problems in the radiotherapy with electrons (

Equipped with the energy deposition methods for electrons in the MeV domain

we now pass to the bremsstrahlung creation of the differences between a conventional tungsten target and a novel target design according to

The thickness of each plate amounts to 0.01 mm, and the tungsten plates are surrounded by a tungsten wall and a ferromagnet.

The distance between the plates (in vacuum) is 1 mm, but the above construction can easier be realized, if the thickness of each plate is 0.02 mm and the distance between the plates 2 mm.

The solution function of Equations (7a, 8) and the energy eigen-values are given by:

ψ k = ∑ n = 0 ∞ 1 n k ! ⋅ ( b k + ) n k ⋅ e − i E n k t / ℏ ⋅ e i ⋅ n k ⋅ ω ⋅ t ⋅ ψ k , 0 ; b k ⋅ ψ k , 0 = 0 ; ( b k + ) n k ⋅ ψ k , 0 = ψ k , n ( k = 1 , 2 ) } (12)

The general solution of Equation (12) is given by:

ψ k = ∑ n k = 0 ∞ 1 n k ! ⋅ e − i E n k t / ℏ ⋅ e i ⋅ n ⋅ k ω ⋅ t ⋅ [ ψ n k − ρ k ⋅ n k ⋅ ( n k − 1 ) ⋅ ψ n k − 2 ] (13)

E n k = ℏ ⋅ ω k ( n k + 1 2 ) + n k ⋅ ℏ ⋅ ω − n k ⋅ ρ k (14)

Since the incoming electron beam has a Gaussian distribution (see also the appendix) we make use of the fixations: The field length at entrance is as usual 4 mm, the height of the 6 MeV Gaussian distribution at the boundaries amounts to 6/e^{2} = 0.135335 = 0.8120117 MeV. Therefore, we have to choose e_{0}·U_{1} = 6 MeV and e_{0}·U_{2} = 0.8120117 MeV, i.e., e_{0}·U_{11} = 6. 8120117 MeV and e_{0}·U_{22} = 5.1879883 MeV. The transition probabilities related to the yield of the bremsstrahlung spectra is received a modified version of Equation (9c) for the absorption and emission processes:

Thus, W_{kl} is related to the transition probability and P_{k} to the probability of the state with index k. The evaluation of the induced transitions with help of Formulas (12) - (14) is rather manageable by the harmonic oscillator state functions. With regard to the energy deposition of the incident electrons we have assumed further assumptions: For tungsten (W) we assume Z = 74 and A_{N} = 183.840; μ ⋅ c 2 is the electron rest energy with reduced electron mass, which can be replaced by m_{el}·c^{2}. The ionization potentials I_{ionp}_{,11} and I_{ionp}_{,22} are 6.812 MeV and 5.188 MeV, respectively, and the ground state energy E_{0} is assumed to be the electron rest energy. The density of tungsten serves as the reference value, δ_{ref} = 19.25 g/cm^{3}. By that, the multitarget provides the ratio δ/δ_{ref} = 0.48125/19.25 = 0.025, E_{r} represents the remaining electron energy, which is neither absorbed to produce dose deposition (heat) nor to create bremsstrahlung above 15 keV. The index is either p = s, if U_{s} = U_{1} + U_{2} or p = d, if U_{d}= U_{1} − U_{2}, and I_{ionp}_{,p} as well as E_{n}_{,p} refers to the same index “p” (I_{ionp,p}: I_{ionp}_{,11} or I_{ionp,}_{22}, E_{n}_{,p}: E_{n}_{,11} or E_{n}_{,}_{22}). It should also be pointed out that the zero point of I_{ionp}_{,p} has been shifted to the harmonic oscillator ground state energy, since this item remains unchanged and, therefore, Formula (14) could be slightly simplified to avoid too long formulas, in particular with regard to Formula (15).

E n , p = I i o n p , p − ( 1 − e − n ⋅ Z ⋅ e 0 ⋅ U p ⋅ δ A N ⋅ μ ⋅ c 2 ⋅ δ r e f ) ⋅ I i o n p , p (14a)

Therefore, the principal question is, what are the differences to the case with one single uncoupled circuit. The most interesting property of Equations (12) - (14) result from the choice of U_{1} and U_{2}, and for this purpose, we consider the creation of 6 MV bremsstrahlung based on two different model. The use of coupled oscillators enables to simulate the energy spectrum of the incident electrons. Formula (14a) reflects the idealistic situation that the total energy of the incident electrons would be converted to bremsstrahlung.

However,

therefore, the dose deposition in the length of 1 mm amounts between 25% and 40% of the incident energy. The third part of energy loss of the electrons by passing through the multitarget results from those electrons, which could not be attributed to small angle scatter and, by that, a further possibility to create bremsstrahlung is prevented. These three influences do not render to put λ = 1, which would yield the creation of bremsstrahlung by the complete energy of the incident electrons, but λ = 0.7071 turned out that the spectral distribution of

E n , p = E r + λ ⋅ [ I i o n p , p − ( 1 − e − n ⋅ Z ⋅ e 0 ⋅ U p ⋅ δ A N ⋅ μ ⋅ c 2 ⋅ δ r e f ) ⋅ I i o n p , p ] + 1 − λ 2 ⋅ i ⋅ [ I i o n p , p − ( 1 − e − n ⋅ Z ⋅ e 0 ⋅ U p ⋅ δ A N ⋅ μ ⋅ c 2 ⋅ δ r e f ) ⋅ I i o n p , p ] (15)

With regard to the standard target according to _{ref} = 19.25. The thickness for the copper plate amounts to 1 mm, too, used to control the temperature of the tungsten target; the effective atomic charge and effective atomic number are: Z_{eff} = 76.4 and A_{N}_{,eff} = 194.5 and λ has to be replaced by 0.438 → (1 – λ^{2})^{1/2} = 0.899. By using these modifications Equation (14a) reads:

E n , p = E r + 0.438 ⋅ [ I i o n p , p − ( 1 − e − n ⋅ Z e f f ⋅ e 0 ⋅ U p A N , e f f ⋅ μ ⋅ c 2 ) ⋅ I i o n p , p ] + 0.899 ⋅ i ⋅ [ I i o n p , p − ( 1 − e − n ⋅ Z e f f ⋅ e 0 ⋅ U p A N , e f f ⋅ μ ⋅ c 2 ) ⋅ I i o n p , p ] (15a)

Thus, E_{n,p} now contains a real and an imaginary part, the latter contribution is connected to losses described above and can be associated to a heat reservoir (energy dissipation). _{average} = 4.00798418482664 MeV, E_{maximim} at 2.98 MeV.

E n , p = E r + λ ⋅ [ I i o n p , p − ( 1 − e − n ⋅ Z ⋅ e 0 ⋅ U p ⋅ δ A N ⋅ μ ⋅ c 2 ⋅ δ r e f ) ⋅ I i o n p , p ] + 1 − λ 2 ⋅ i ⋅ [ I i o n p , p − ( 1 − e − n ⋅ Z ⋅ e 0 ⋅ U p ⋅ δ A N ⋅ μ ⋅ c 2 ⋅ δ r e f ) ⋅ I i o n p , p ] (15b)

The imaginary part of the above equations reflects those losses, which cannot be transmitted to bremsstrahlung. This is, above all, the production of heat in the target system and scatter. The heat production is accounted for by the function E(z) according to Equation (11), whereas the influence of scatter (photons and electrons) is handled by Gaussian kernels [

P ˙ k = ∑ l = 1 ∞ ( W k l P l − P k W l k ) (15c)

Thus, W_{kl} is related to the transition probability and P_{k} to the occupation probability of the state with index k. The evaluation of the induced transitions with help of Formulas (12) - (15c) is rather manageable by the harmonic oscillator state functions. With E_{0} = 6 MeV in Formula (29) only one specific incident electron energy is assumed. The parameters of Equation (16) are given in

Standard case | Multitarget focused by a magnetic lens | |
---|---|---|

p | 1.436 | 1.5591 |

q | 0.405 | 0.4048 |

β | 1.055 | 1.0493 |

α | 9.321 | 7.1102 |

used for the spectral representation of other usual energies in radiotherapy, e.g., 10 MV, 15 MV, 18 MV, but also in the X-ray diagnostics (CT, CBCT, etc.). Formula (16) results from the Laplace transform of depth dose curves, providing the possibility to determine the energy spectrum, i.e., the bremsstrahlung is stated in MV instead of MeV, by experimental data received in dosimetry [

f ( E ) = ( 1 − e − α ⋅ E E 0 ) p ⋅ e − β ⋅ E 2 E 0 2 ⋅ ( 1 − E E 0 ) q E average = ∫ 0 E 0 f ( E ) ⋅ E ⋅ d E / ∫ 0 E 0 f ( E ) ⋅ d E } (16)

A further restriction of the choice of the target thickness is the radiation length stated in

The average energy E_{average} according to Equation (16) amounts to 2.4057 MeV for the standard case and 2.795 MeV for the multitarget case. In

The basic physical processes occurring in the target according to the previous

This Figureshows the close connection between bremsstrahlung creation and scatter (in particular, the scatter in a dense medium like tungsten). Since

scattered electrons have to cover a longer distance, which is usually referred to as detour factor, the creation of bremsstrahlung may either be suppressed by heat production or leave the target with a large angle; this behavior is shown by the dashed curve in

Nucleus | E_{c} (calculation)/MeV | E_{c} (formula of thumb)/MeV | R_{l}/mm | X_{0}/g·cm^{−2 } |
---|---|---|---|---|

Al | 42.6230 | 42.837 | 89.924 | 24.27948 |

Fe | 22.4367 | 22.3935 | 17.9550 | 14.13058 |

Cu | 20.0623 | 20.1720 | 14.7577 | 13.16387 |

Pb | 8.0216 | 7.3282 | 5.5602 | 6.30527 |

W | 7.9613 | 8.1074 | 3.5151 | 6.76657 |

Au | 7.5892 | 7.6022 | 3.3263 | 6.42641 |

Water | 48.04 | 48.426 | 360.80 | 360.80 |

In

An interesting aspect is the creation of crossing radiation beams of photons with the energy E_{photon} ≥ rest energy of electrons (0.521 MeV). In a fundamental paper dealing with photon scatter based on the Dirac equation [

The configuration at the right-hand side seems to be more suitable to realize the task because of the size of the apparatuses. In both figures, we have omitted that the low-energy electrons leaving at the ends of the multi-target channels must be deflected by additional magnets in order to prevent disturbing effects at the detector arrays. The Heisenberg-Euler scatter is not a first-order effect in relativistic quantum theory.

A significant improvement of the present configuration of _{0} = 9 MeV we also get significant contributions in the domain between 4 and 8 MeV, which certainly enhances the measurement conditions.

The magnetic field strength leading to

A look on Formula (14a) shows that for energies in the keV domain, where the voltage is of the order of about 120 kV, e_{0} ∙ U_{p} is much smaller than the electron rest energy μ c 2 ≅ m electron ⋅ c 2 . Therefore, the essential term in the exponential function is only of first order and terms of higher order than Z^{1} only provide small corrections (this aspect is also analyzed in Section 3.1). This fact provides an interesting proposal with regard to the CT-scanning methods: An attractive version of the cited scanning machine in the MeV domain [

In _{av} amounts to E_{av} = 57.68 keV. The dashed curve is the result of the multitarget described above with 10 layers (thickness 0.01 mm, distance between 2 layers: 5 mm). The magnetic field strength is assumed to be 0.75 Tesla. The maximum of the bremsstrahlung amounts to E = 44.67 keV and the related average energy to E_{av}= 64.67 keV. The characteristic peaks of tungsten resulting from shell transitions can be verified in both cases. It has to be mentioned that discrete interval steps would only be possible in the dashed curve referring to the multitarget with a diameter of 4 mm, whereas the solid curve results from a different technique, namely the reflection at a rotating tungsten disk. The scoring plane is immediately the disk. If one traces the produced beam up to the jaws, a divergent broad beam can be recognized, since the impinging electron beam at the tungsten disk is oblige. Therefore, a discrete representation of relative energy fluence does not make sense due to the lack of comparability.

The photon-photon scatter originally considered by Heisenberg & Euler seems to an indication that with very narrow and, by that, high intensity beams should

be extended to the GeV domain. However, this task may imply some performance difficulties, namely to create two crossing beams in this energy domain. Therefore, the main motivation to make use of modifications of a standard target according to

Keeping the physical background of

present multitarget configuration with tungsten wall and magnetic field can be verified in

The spectral distribution related to ^{2})^{1/2} = 0.586982.

As already pointed out in the previous section (Heisenberg-Euler scatter), the basis of

The problem of the bremsstrahlung creation concentrated to small field sizes with relevance to novel irradiation techniques has already studied in ref. [

Section 2 shows that coupled circuits can find applications in different energy regions of physics. Thus, in various physical disciplines the harmonic oscillator approach with constant differences between the energy levels may not be suitable to describe actual problems. There are two different ways to overcome this problem: 1) One may use harmonic oscillator conformations and take account for the polarizability of the dielectric constant in dependence of the energy under consideration. This behavior is increasing with increasing energy. By that, the whole task turns out to be a nonlinear one. 2) A further access to overcome this problem uses nonlinear field theory either in position space [

The essential term of Formulas (13) - (15) is related to the exponential energy spectrum creation of bremsstrahlung resulting from mutual couplings (M and C_{I}) of the circuits according to _{0}·U_{p} − E_{av} (average energy of the target material, which is for tungsten E_{av} = 74.45 eV, see Section 2). For the following discussion the basic equation (14a) is repeated here:

E n , p = I i o n p , p − ( 1 − e − n ⋅ Z ⋅ e 0 ⋅ U p ⋅ δ A N ⋅ μ ⋅ c 2 ⋅ δ r e f ) ⋅ I i o n p , p (17)

The power expansion of the parenthesis of this formula can be used to provide further results:

− ( 1 − e − n ⋅ Z ⋅ e 0 ⋅ U p ⋅ δ A N ⋅ μ ⋅ c 2 ⋅ δ r e f ) = − [ 1 + n ⋅ Z ⋅ e 0 ⋅ U p ⋅ δ A N ⋅ μ ⋅ c 2 ⋅ δ r e f − 1 2 ( n ⋅ Z ⋅ e 0 ⋅ U p ⋅ δ A N ⋅ μ ⋅ c 2 ⋅ δ r e f ) 2 + ⋯ ] (17a)

The connection of the expansion (17a) with Formula (17) shows that for energies in the keV domain, where the voltage is of the order 100 kV, e_{0} ∙ U_{p} is much smaller than the electron rest energy μ c 2 ≅ m electron ⋅ c 2 . This property has been used in Section 2.2.3, in which we have described an application to CT methods. Therefore, the terms of higher order than Z^{1} only provide extremely small corrections.

On the other hand, the higher order terms Z^{2}, Z^{3}, …, become significant with increasing electron energy in the MeV domain. By that, we can conclude that the developed formalism works in the keV as well as in the MeV domain, and the main effort is the evaluation of the transition probabilities. In the following Section 5 we present some indication that the elaborated methods also work in the GeV-domain, since the exponential expression already includes all terms of higher order, and the multitarget according to

R c s d a = [ ∑ k = 1 4 a k ⋅ E 0 k + a 5 ⋅ E 0 p ] ⋅ C f a 1 = 0.4092031 ; a 2 = − 9.80895 × 10 − 4 ; a 3 = 1.16056 × 10 − 6 ; a 4 = − 4.95957 × 10 − 10 ; a 5 = 0.67168125 ; p = 9.0875 × 10 − 2 E 0 = b 1 ⋅ R c s d a ⋅ e μ ⋅ R c s d a [ 1 − b 2 ⋅ R c s d a q ] ⋅ C f b 1 = 13.1503746738472 ; b 2 = 0.787625044162269 ; μ = 2.331100073934067 × 10 − 2 ; q = 3.527749515160394 × 10 − 2 } (18)

The present calculation formulas for the Rcsda range and related energy as the inverse task represent slight modifications of the previous formulas valid up 30 MeV, valid for the energy range > 30 MeV. If compared with ICRU [_{csda} → R_{csda} – z has to be performed. However, we should finally point out that only Formula (10) - (11) are recommended for the energy domain of interest in the radiotherapy with electrons.

A noteworthy feature emerges by the role of the voltages, which may either be time-dependent U_{1}·e^{iωt} and U_{2}·e^{iωt} or static by setting ω = 0. Due to the mutual coupling of the circuits according to _{11} = U_{1} + U_{2} and U_{22} = U_{1} − U_{2}. In the present study we have fixed: U_{1} = 6 MeV, U_{2} = 0.812 MeV; U_{1} = 9 MeV, U_{2} = 0.95 MeV; U_{1} = 9 GeV, U_{2} = 0.7 GeV. By that, we obtain: U_{11} = 6.812 MeV, U_{22} = 5.188 MeV; U_{1}_{1} = 10.218 MeV, U_{22} = 7.782 MeV; U_{11} = 9.7 GeV, U_{22} = 8.3 GeV (in the latter case the numerical values are estimated, since the Gaussian distribution may probably be invalid). These assumptions may adapt the Gaussian distributions of the incident electron current. Since due to the mutual couplings only the sums and differences of U_{1} and U_{2} enter the calculation procedures of the energy spectra, we receive much higher flexibility, e.g., by putting U_{1} = U_{2}, the sum U_{11} = 2 · U_{1}, whereas U_{22} assumes 0. This means that one normal mode is not coupled to a force!

A further aspect of the received results is a side-effect, the shielding problem with respect to individual cover plates in radiotherapy of electrons. Thus, the above Formula (10a) can be used for the calculation of the energy loss of a lead alloy. On the other hand, the creation of bremsstrahlung in this alloy is a severe problem, insofar the individual cover plate does not exhibit the sufficient thickness. If one uses 4 MeV or 6 MeV electrons suitable for subcutaneous irradiation a shielding thickness of 2 cm would be acceptable, but the application of electron energies beyond 6 MeV must account for increasing thickness of the shielding alloy else the undesired irradiation with bremsstrahlung produced in the shielding material would occur.

The quantum theoretical treatment of circuits implies similar as in the quantum mechanical case equidistant energy levels, if the material properties of the capacitor do not depend on the energy domain. This assumption is, in general, not valid, and the relation C = ε · C_{0} has to be replaced by C = ε(ω) · C_{0}, i.e., the dielectric constant ε depends on the frequency of energy due to the polarization interaction. Thus, if electrons interact with numerous electrons in a material, polarization effects will always consequently be evoked. This behavior is not only restricted to existing electrons, e.g., in a metal like tungsten, but also important for fast electrons traveling in vacuum by inducing the so-called “vacuum polarization” by interaction with virtual electrons and positrons [

In this study, we have made use of the polarization effects in the capacitors in order to account for the induced virtual states necessary at the creation of bremsstrahlung. An improvement and refinement of the presented conception would be the use of three (or more) mutually coupled circuits. However, it appears that in the MeV domain with predominance in radiotherapy this extension is not required. Therefore, only the bremsstrahlung production in the high energy physics could be described by a refined conception, in particular, if one passes to electron energies beyond the domain of 9 GeV. Since the particular view of this study are quantized circuits, the results are restricted to distributions in the energy space, and the spatial distributions are ignored. However, this shortcoming has been removed by further information resulting from Monte-Carlo calculations with the code GEANT 4. It should also be added that it is certainly attractive to study the bremsstrahlung production besides 9 GeV of further ultra-high energies in order to elucidate quark and gluon properties by a deeper level, since the LPM-effect can be reduced drastically by the wall reflection in a multitarget system and the focusing influence of magnetic fields. Since uranium, which is also paramagnetic, might possibly more suitable for these studies, a substitution of tungsten by uranium could provide better results. The theoretical procedure developed in Section 2 has not to be extended.

The introductory part has provided information about the wide field of quantized circuit models, such as quantum optics, Josephson junctions in superconductivity, molecular biophysics and physical basis of biorhythms. It appears that this field of quantized circuits may be completed by additional references [

The author declares no conflicts of interest regarding the publication of this paper.

Ulmer, W. (2021) The Role of Forced Oscillators of Coupled Circuits in Radiation Physics: New Linear Accelerator Design Improving Tomo-Scanning Technology (Radiotherapy and CT), Heisenberg-Euler Scatter, and Extension to Bremsstrahlung with GeV Electrons. Journal of Applied Mathematics and Physics, 9, 707-735. https://doi.org/10.4236/jamp.2021.94051

Formulas (19), (20) have been developed by the present author with regard to some problems of describing the electron beams in Monte-Carlo calculations with GEANT 4 [

Definitions: The energy E referring to the spectral distribution and the nominal energy E_{0} only refer to the kinetic energy of electrons, their total energy results by the addition of the rest energy m·c^{2}. With regard to the following formulas, it must be noted that all terms referring to energy are considered as dimensionless numerical values, i.e., they are divided by a unit energy E_{unit} = 1 MeV. The following terms are used in order to obtain the energy spectral distribution E_{sp}:

δ = E 0 − E ; s = δ 2 ⋅ π ; σ = π 2 ⋅ [ 1 + erf ( s ) ] erf ( s ) = 1 π ∫ 0 s e − s 2 d s } (19)

E s p = erf ( E / m c 2 ) ⋅ e − δ 2 σ 2 E s p = E s p / Max ( E s p ) } (20)

With the help of the operation E_{sp}/Max(E_{sp}) the maximum is normalized.

Formulas (19) and (20) and

It should be pointed out that the parameters s, σ and δ in Equation (19) depend on the actual energy E, and, by that, the form of the distribution function is determined. Thereafter, the electron beam is bypassed by a bending magnet, and a slit only permits a narrow Gaussian distribution before it impinges the target.